Physics 1

Q34. a = ? Derive the formula for acceleration down an inclined plane.

a = gsinθ

Newton's Third Law:

Action-Reaction Whenever one object exerts a force (via contact or via a field) on a second object, the second object always exerts an equal and opposite force on the first object.

Resolving Vectors into Components:

You must pay attention to the angle. If measured with respect to (w.r.t.) the x-axis, Vcosθ gives the x-component and Vsinθ gives the y-component (where V is the magnitude of the original vector); if measured w.r.t. the y-axis, however, Vsinθ gives the x-component and Vcosθ gives the y-component.

Q9.The moon is approximately 400 times smaller than the sun.If the sun exerts a gravitational force of F on the moon, what is the approximate force exerted on the sun by the moon, in terms of F? (Assume equal densities)

F; these two forces represent an action-reaction pair, so Newton's Third Law says that the magnitudes must be the same.

Q19. If a projectile has an initial velocity of 60m/s at an angle of 30 degrees from the horizontal, how many seconds will it be in the air?

First, find the vertical component, which is all we care about when determining time in the air. The y-component is given by 60(sin30) = 60*0.5 = 30 m/s. This projectile will take 3 seconds to reach max height and 3 more seconds to reach the ground, for a total of 6 seconds.

Q20. If a projectile has a vertical velocity of 100 m/s, how many seconds will it take to reach max height and what will be its average velocity?

It will take 10 seconds to reach max height. Its average velocity will be 50 m/s.

Terminal Velocity:

o At terminal velocity mg = Fair. At terminal velocity, the object has stopped accelerating; the forces of gravity and air resistance are now balanced.

Q6.Track the velocity of a ball thrown by a pitcher—from the moment the pitcher begins his pitching motion until the ball hits the ground (no catcher, ignoring air resistance).

(This assumes the pitcher throws the ball perfectly horizontally, with no vertical component.) Ignoring air resistance, the ball starts at rest in the pitcher's hand. The ball experiences a force created by the pitcher that accelerates it forward. This acceleration only exists during the time that the pitcher's hand (the source of the force) is in actual contact with the ball. Once the ball is released it becomes a projectile in free fall. It will maintain the exact horizontal velocity it had at the moment of release throughout its flight. Gravity will accelerate the ball toward the earth at 9.8m/s2 in the vertical direction only. There will be no acceleration in the horizontal direction and therefore horizontal velocity will be constant. When the ball strikes the ground it will have the vertical component of final velocity created by its fall from the height of release to the ground, plus the same constant horizontal velocity it had throughout the flight. NOTE that it will be in the air for the same amount of time as it would have been had the pitcher dropped it from the same height and let it fall directly to his feet.

Projectiles:

(Which are nothing more than falling body problems in two dimensions) o Immediately resolve the given vectors into components and then solve in your head as you would any other motion problem. o "Range," or horizontal distance traveled, is a new value asked for in projectile motion. Range = Vx*time o When you see projectiles THINK: 1) Horizontal velocity never changes (ignoring air resistance). 2) Horizontal acceleration always = 0. 3) Vertical acceleration always = 10 m/s2 downward. 4) Vertical behavior is exactly symmetrical (i.e., the upward trip is identical to the downward trip). 5) Time in the air depends on the vertical component of velocity only. 6) Range depends on both the vertical and horizontal components. 7) Time is always the same for both the x and y components of the motion.

Uniform Acceleration, Projectiles and Free Falling Bodies Steps to solving motion problems in your head:

1) Conceptually define acceleration: Acceleration is the change in velocity/second; or the change in m/s each second; or the "rate of change" of velocity. Because we know how velocity changes for all bodies under the influence of earth's gravity (10m/s each second; ignoring air resistance), we can predict velocity at any time period. If a ball is going up, for example, it loses 10m/s of velocity each second. If a ball is falling it gains velocity at 10m/s each second. 2) Learn how to calculate average velocity quickly. To find average velocity, simply take the initial velocity and add it to the final velocity, then divide by two (only consider the upward half or downward half of the motion and then one of them will always be zero). Vavg = (V1 + V2)/2. 3) Calculate the distance (or height) traveled using Distance = rate*time. In steps 1) and 2) above, you just calculated the average velocity (rate) and the time. Multiply these together and you'll know exactly how high the object travels.

The difference between the actual weight and the apparent weight tells you:

1) The buoyant force and 2) the weight of that volume of fluid. For example, suppose you are told that the density of a fluid is 2.0 g/mL and that a 5 kg object fully submerged in that fluid has an apparent weight of 40 N. The actual weight is 50 N. When we subtract that apparent weight of 40 N we get 10N. This is the buoyant force. More importantly, it tells us how much one "objects-worth" of fluid weighs. We already stated that the object weights 50 N. Since it is fully submerged, that exact same volume of fluid weighs 10 N. If that exact same volume of fluid weighs only 10 N, then we know that the fluid must be 1/5 as dense as the object.

Q18. If a projectile has an initial vertical velocity of 30 m/s, how many seconds will it take to reach its max height?

3 seconds.

v = √(2gh) Q24. You can use the above equation to calculate either initial or final vertical velocity. Why?

Because initial velocity and final vertical velocity are the same for any projectile due to the symmetry of projectile motion. Of course, this is only for objects leaving the ground and returning to earth. Obviously, if the motion starts from rest, say at the top of a cliff or building, then the initial velocity would be zero. That would be known or inferred and wouldn't need to be calculated.

Q36.As the angle of incline of a plane increases, what happens to the value of a?What happens to the value of sinθ and cosθ? What happens to the normal force and to the force down the plane? What are the minimum and maximum values for acceleration down an inclined plane?

Because the acceleration down a plane is directly related to the sine of the angle, the greater the angle the closer the sine of the angle will be to one. Therefore, the larger the angle the closer the acceleration will be to 9.8m/s2. The normal force is related to the cosine of the angle, so as the angle increase this value gets closer to zero. Therefore, as the angle increases the normal force decreases. The force down an inclined plane is also related to the sine of the angle, so it too will increase as the angle of incline increases. The theoretical maximum incline is 90 degrees where acceleration would be 9.8m/s^2 and the minimum would be a plane with no angle of incline, where acceleration down the plane would be zero.

How does the buoyant force change with depth? How does the buoyant force change with the mass of the object?

Buoyant force does not change with depth! This is because the change in depth between the top and bottom of the object does not depend on depth. Remind students that one of the best ways to predict how (or if) an entity is altered by a change to one specific variable is to look to see if the variable in question is in the formula for that entity. This can not only tell you if it does or does not have an impact on the value of the entity, but can tell you if that influence is linear, exponential, etc. In this case, depth (h) is not in the formula: F = pvg. Therefore, we can confidently conclude that depth does not change buoyant force. The conceptual approach we just described for understanding buoyant force would help us as well—whether the depth is shallow or very large, the pressure difference between pgh-top and pgh-bottom will remain the same. Similarly, we can say that the mass of the object does not affect buoyant force because it is not in the formula, nor accounted for by our understanding of what causes the buoyant force.

Vectors and Scalars

Definition: Vectors have both magnitude and direction. Scalars have only magnitude.

Q11. How does distance change as a car drives around a circular track? How does displacement change? Graph both distance and displacement vs. time for the car.

Circular motion is a good illustration of the dramatic difference between distance and displacement. A racecar driver on a circular path has a constantly increasing distance traveled but a cyclical displacement that is zero at regular intervals. The graph of distance vs. time would be linear if speed is constant. The graph of displacement vs. time would approximate a sine wave with the minima equal to zero.

Friction

Conceptual Understanding: REMEMBER: Friction opposes sliding, NOT motion Kinetic Friction vs. Static Friction: If there's sliding, it's kinetic friction; if there's no sliding, it's static friction. Formulas: o Ff=usFN or Ff=usmgcosθ o Ff=ukFN or Ff=ukmgcosθ Max Static Friction: In cases of static friction, the friction created before an object begins to slide will always remain equal to the net applied force which the friction is opposing. For example, if you push on a boulder with 20N of force there will be 20N of static frictional force opposing you. If you increase the force you apply to 100N the static friction will also increase to 100N. This continues up to the "maximum static friction." Once this value is exceeded the object will begin to slide and we then have kinetic friction, not static.

Gravity

Definition: Gravity is a field that exists between any two objects with mass. o THINK OF A FIELD AS: Field = an invisible influence capable of exerting a force on a mass or charge (The "charge" part will make more sense after we study electric fields).

Q10. Distinguish between the following: distance and displacement.

Distance is a measurement of travel along any path, curved, linear or otherwise. Displacement is a vector indicating the change in position from start to end. The magnitude of the displacement is the shortest distance between the starting and ending points. Distance is DEPENDENT on path. Displacement is path INDEPENDENT.

Manipulating Equations

Equations: For the purpose of manipulating equations and seeing relationships only , know these three equations: 1) x = 1/2at^2 2) v = √(2gh) [Which can also be written V = √(2ax)]. This is the only equation we'll allow you to plug and chug. Use it when asked for final velocity given drop height. You can also use it as a stepping stone to find the final velocity. Once you know final velocity you can intuit all other parameters of the motion using the same conceptual steps given previously. 3) t air= 2v/g (Used only to calculate "round trip" times, or in other words, the total time in the air. The variable V must be the vertical component of initial velocity).

Q8. Provide at least three examples of action-reaction pairs.

Examples of action-reaction pairs: any contact force between two objects; gravitational force between any two objects (regardless of mass); electrostatic force between two charged particles (regardless of charge); magnetic force between two objects; plus many other possible scenarios.

Buoyant Force

F buoyant = pVg Where V is the volume of fluid displaced, NOT the total volume of the fluid, and d is the p is the density of the fluid, NOT the object.

Q32. Which way does the friction force vector point for a car driving east down a straight road? For a skidding car with locked brakes? For a gecko climbing a wall? For a car driving around a corner?

For a car driving east the friction vector between the road and the tires also points east. The tendency of the tires is to slide past the road toward the back of the car and friction opposes that sliding. When a car locks its wheels, however, friction points opposite the direction of the car's tendency to slide forward. Friction would point up the wall for a gecko because the geckos hands would tend to slide down the wall. For a car going around a bend momentum makes it want to continue tangent to the circle but friction points toward the center of that circle keeping it in circular motion (an example of a centripetal force).

The Universal Law of Gravitation

Formula: F = Gm1m2/r2 o Often referred to as the "inverse square law" because force varies inversely with the square of the distance r. G, the gravitational constant, is 6.67 x 10-11 m3/kg*s2, but is always given. o The Universal Law of Gravitation is true everywhere. Near earth, however, we make an assumption that gravity is a constant 10 m/s2 despite the fact that this law shows that it varies ever so slightly with height. Based on this assumption, we can simplify to: F = mg.

Inclined Planes

Formulas: o F = mgsinθ (Force down an inclined plane, parallel to the surface) o FN = mgcosθ (Normal Force on an inclined plane) o Vf = √(2gh) (Velocity of a particle at the base of an inclined plane)

Q15. If there is no net force, can there be acceleration? If the force increases,what happens to acceleration? If there is no acceleration is, or could there be, a force?

If there is no net force there can never be an acceleration according to Newton's Second Law. However, there CAN be a force and no acceleration if there are other forces to counteract it. If force increases, acceleration increases (all other forces and factors remaining unchanged). If force increases linearly acceleration increases linearly. If force increases exponentially acceleration increases exponentially. If there is no acceleration there could be a force acting on that object. However, if there is a force we would know that it must be exactly canceled out by other force vectors on that same object (the object must be in equilibrium).

Air Resistance

Important Characteristics: o Air resistance is the force exerted on projectiles or falling bodies due to actual physical collisions with air molecules. o The following factors affect the magnitude of air resistance: 1) Cross-section Area: greater cross-section area impacting the air = more air resistance 2) Shape: less aerodynamic = more air resistance 3) Velocity: increased velocity = more air resistance. ********Always assume air resistance is being ignored, unless it specifically states otherwise in the question stem or passage!!!!!!!*************

Q21. If a projectile has an initial vertical velocity of 30 m/s, how long will it be in the air, how high will it go, and what will be its average velocity during the entire trip?

It will take 3 seconds to reach max height, and will be in the air for six seconds total. It will travel an average velocity of 15m/s during the upward trip for 3 seconds, which means max height will be 45m. Its average velocity for the return trip will be equal in magnitude to its average velocity for the upward trip, but opposite in direction, so the average velocity during the entire trip will be 0m/s. Alternately, since average velocity is displacement/time, and the displacement of the complete trip is zero, the average velocity is also zero.

Q33. 500N is applied to an object and it does not move. 501N is applied and it just begins to slide. Describe the amount of force that must be applied to the object continuously to move it at a constant velocity across the surface.

Kinetic friction is always less than static friction for the same two surfaces. To maintain constant velocity the applied force must exactly counterbalance the kinetic friction. Therefore, we would expect that some force less than 501N will be required. If the same force of 501N remains on the object we can predict that it will accelerate.

Adding, Subtracting and Multiplying vectors:

Know how to add or subtract vectors both graphically (head to tail method) and mathematically (component method). Subtracting Vector B from Vector A is identical to adding the negative of Vector B to Vector A. When multiplying two vectors, if the answer is a scalar (e.g., work) you must also multiply by cosθ (e.g., W = Fdcosθ; a.k.a. the "scalar" or "dot" product), where theta is the angle between the two vectors. If you are multiplying two vectors and the answer is a third vector (e.g., torque), you must also multiply by sinθ (e.g., T = Frsinθ; a.k.a. the "vector" or "cross" product).

Q5. The law of inertia states that____?

Law of Inertia: An object in motion tends to stay in motion (in the same direction and at the same speed), and an object at rest tends to stay at rest, unless acted upon by some net external force. A very simple way to put it would be: Objects with a constant velocity maintain a constant velocity unless acted upon by a net outside force. This takes care of both "at rest" and "in motion" and any questions about changing direction because a change in direction is a change in velocity.

Q25. Describe two sky divers jumping from a plane, one of mass 100 kg and one of mass 150 kg. Describe all vectors affecting them at all times, when and how each jumper reaches terminal velocity, how their terminal velocities compare, who hits the ground first, and why. Assume both have the same cross-section area.

Let's call the 100kg skydiver Jeff. Let's call the 150kg skydiver Jack. The moment they drop out of the airplane Jack experiences a larger force due to gravity. The force experienced due to gravity by either skydiver is simply mg. The value of mg will be larger for Jack because his mass is larger. If they have approximately the same surface area they will feel about the same force due to air resistance at the same velocity (air resistance is directly related to both velocity and surface area). Even within the first few seconds of falling, however, Jack will already be traveling at a higher velocity than will Jeff. This is because if they both experience about the same force due to air resistance that force will constitute a larger fraction of Jeff's gravitational force than it will of Jack's gravitational force. Their individual accelerations will be given by: a = (Fgravity - Fair)/m. The ratio of F/m will no longer be a constant for both skydivers, as it would be if there were no air resistance. Jack will not only accelerate faster, but he will also keep accelerating for a much longer time than Jeff. Also notice that while without air resistance acceleration is always a constant 9.8m/s2, with air resistance acceleration will continually be changing (decreasing, eventually reaching zero at terminal velocity). This is due to the fact that F air increases with increasing velocity. At whatever point Jeff reaches terminal velocity (when his mg = F air), Jack will NOT be at terminal velocity yet because he requires a bigger force due to air resistance to balance out his larger mg. When Jack does reach terminal velocity the magnitude of his terminal velocity will be larger than Jeff's. At terminal velocity both will be in equilibrium, but Jack will be experiencing the larger mg AND the larger F air. For all of these reasons Jack will hit the ground first. NOTE to students clearly that if there was no air resistance Jack and Jeff would hit the ground at the same time. For either skydiver, a = F/m, or a = mg/m when the force due to gravity is the only force acting. The masses cancel showing that the acceleration is g in both cases. Another way to think of it is that Jack's force due to gravity (mg) is multiplied by a factor of 3/2 due to his larger mass, but because Jack's mass is also in the denominator of the formula a = F/m, the denominator will also increase by 3/2—so there is no net effect. Absent air resistance the ratio of F/m is a constant for objects of any mass and is equal to 9.8 m/s2.

Linear Motion Acceleration:

Make sure you understand acceleration conceptually. Many students' natural, intuitive sense of acceleration is that it describes how fast you are going, or that something that is accelerating is going "really, really fast." Actually, that's not true at all. Acceleration describes how quickly the velocity is changing, not the velocity itself. Going from zero to 60mph is acceleration, but so is going from 0.00001 m/s to 0.000011 m/s. For that matter, going from 5 m/s to 4 m/s is also acceleration. How big is an acceleration of 10 m/s^2? This is the approximate acceleration of gravity and the one acceleration with which we probably have the most natural experience. If you were to drop a rock from a 15-story building—about the size of a hotel tower or small downtown office building—it would take about 3 seconds to hit the ground and would reach a final velocity of around 30 m/s, or 67 mph. That is almost exactly equal to the maximum acceleration of a Lamborghini Murcielago, one of the world's fastest production sports cars in terms of 0 to 60mph acceleration.

Mass

Mass is a measure of an objects resistance to change in velocity (so it is a measure of inertia). Conceptually, you can think of it as how much matter or stuff makes up the object (although this stuff concept starts to break down when we start thinking on a subatomic scale while "a measure of inertia" holds up).

Is it possible for Apparent Weight to be greater than Actual Weight.

No. Apparent weight is accounted for by the fact that the actual weight of the object is reduced by the magnitude of the buoyant force. It only applies to objects more dense than the fluid. If the object were less dense than the fluid it would float. When an object is floating, mg (actual weight) = buoyant force, so by definition the apparent weight must be zero. To see this more clearly, rearrange the equation to: F buoyant + Apparent Weight = Actual Weight.

Q12. Can you use the formula D = rate*time to calculate displacement? Why or why not?

No. As seen in the above example of the racetrack, multiplying rate times time gives distance but not displacement. Technically, if all motion was in a straight line you could use this formula. In that special case distance and displacement would have equal magnitude.

Q26. What will be the terminal velocity of an object falling in a vacuum?

Objects in a vacuum do not attain a terminal velocity because there is no air resistance. They continually accelerate.

Linear Motion Velocity:

On the MCAT, you can treat "speed" the same as "velocity" if (and only if) the question makes it clear that the distance traveled is along a straight line. When this is the case, the magnitude of distance = magnitude of displacement and the magnitude of speed = the magnitude of velocity. The MCAT authors seem to have a small penchant for using terms like "constant speed" instead of "constant velocity," so don't let that surprise or confuse you. Constant velocity: If you were to make a "Top 10 List" of the most frequently tested MCAT principles, this would be near the top! When you see "Constant Velocity" or "Constant Speed" THINK: 1) No acceleration 2) No net force 3) All forces sum to zero (i.e., up forces = down forces, left forces = right forces, etc.) 4) No change in direction 5) The object is in equilibrium

Q31. What causes friction on a microscopic level? On the molecular level?

On the microscopic level even the very smoothest surface is actually extremely rough. The peaks, valleys, protrusions , etc. of the two surfaces literally collide with one another. On the molecular level, the electron clouds of atoms do occupy real space. When we try to get them to occupy the same space or bring them very close together they will repel one another. This is why what we will just call "contact forces" for the MCAT are often classified as an example of electromagnetic forces. This interaction is actually due to repulsion between electrons.

Range depends on both the vertical and horizontal components. Q22. Why? Explain the dependence of range on both the x- and y-components of velocity.

Range depends on both vertical and horizontal velocity because vertical velocity determines how long the object is in flight. So, the longer it is in flight the more time it will have to travel horizontally. Horizontal velocity will determine how far it travels within the time limits put on it by the vertical velocity.

Q27. One projectile is launched in a vacuum and a second projectile is launched in the presence of air resistance. Both projectiles are given identical initial velocities and launch angles. What will be the differences in range, max height and final velocity for the two projectiles?

Range: The projectile experiencing air resistance will have a shorter range for two reasons. First, air resistance will create a force that impedes the horizontal component of the projectile's velocity. The projectile's horizontal velocity will be less and therefore its range will be less. Second, air resistance will also impede the vertical component of the projectile's velocity. This will decrease max height and time in the air. Less time in the air allows for less horizontal travel before hitting the ground. Final velocity will also be less with air resistance because (as already stated) 1) horizontal velocity will be less, 2) vertical velocity will be less, and 3) therefore the total final velocity will be less.

5) In practice, the acceleration due to gravity is not a constant 9.8m/s^2, but varies with distance from the center of the earth. Taking this into consideration, as a falling object approaches the earth it will: A) exhibit uniform acceleration, but its velocity will increase B) exhibit an increasing rate of velocity change C) exhibit a decreasing rate of velocity change D) exhibit uniform acceleration, but its velocity will decrease

Solution: Because g is inversely related to r, as the object approaches the earth the rate of velocity change will increase, making B the best answer. Note that once again this question tests thinking and application, not number crunching (80/20).

2) A loaded spring shoots a ball across a flat table. After the ball is no longer in contact with the spring, the acceleration of the ball will: (Ignore friction) A) increase linearly. B) remain constant. C) increase exponentially. D) decrease linearly.

Solution: Once the ball is released, if no friction or other horizontal forces act on the ball, it will maintain constant velocity. (The vertical forces of gravity and normal force will cancel out so there will be no vertical acceleration.) The ball did accelerate, but only while in contact with the spring. Afterward, the ball has no acceleration: a = 0. Since acceleration remains constant at that zero value, choice B is the best answer.

3) A 10 kg block is sliding across the floor at a constant velocity of 4 m/s. The block experiences a frictional force of 20N. A rope of negligible mass is tied to the block at its center of mass and forms an angle of 45 degrees to the horizontal. What is the tension in the rope? A) 28 N B) 45 N C) 20 N D) 23 N

Solution: The block is not accelerating. Thus, all forces in both x and y directions must sum to zero. We know there is a 20 N frictional force opposing the sliding motion.Thus, there must also be 20N of force createdby the rope in the direction opposite of the friction vector. This force is created by the x -component of the tension in the rope. Because the rope is at a 45θ angle, the y-component of the tension also equals 20N. The total tension is given by the Pythagorean theorem, √(20^2 + 20^2) = 20√2 = 20(1.4) = 28, Answer A.

1) A 1.0 x 10^-12 kg mass is passing through space when it encounters a magnetic field of 1.065 x 10^-3 Teslas. This field exerts a force of 2.5 x 10^-10 N on the particle. What is the acceleration of the particle due to the magnetic field? A) 1 x 10^-15 m/s2 B) 2.5 x 10^2 m/s2 C) 2.6 x 10^-13 m/s2 D) acceleration cannot be calculated without knowing the angle between the velocity vector and the magnetic field

Solution: This question introduces extraneous information (the magnetic field strength in Teslas) and tries to lure the test taker into thinking of a common formula, F = qvBsinθ. In reality, Fnet = ma will suffice where Fnet is the single force given. Solving this for acceleration gives a = F/m or 2.5 x 10^-10/1 x 10^-12, which gives answer B.

Q29. What force does the earth experience due to a falling rock?

The earth experiences the same force that the rock experiences. The difference in mass is just so unfathomably large that the earth would take eons of time to move an observable distance under that tiny force.

4) Two students throw identical balls into the air with identical initial total velocities of 40m/s. Student A throws his ball at a 50° angle with respect to the ground and Student B throws his ball at a 40° angle. In comparison to Student A's ball, Student B's ball will: A) remain in the air for a longer period of time, but travel a smaller horizontal distance B) remain in the air for a shorter period of time, but travel the same horizontal distance C) remain in the air for a longer period of time and travel a greater horizontal distance D) remain in the air for a shorter period of time, but travel a greater horizontal distance

Solution:StudentB'sball will haveagreaterhorizontalcomponentofvelocityandasmallerverticalcomponent.Asmaller vertical velocity always means less time in the air, making answers B and D the only possible choices. The optimum launch angle for maximum range is 45°. Because the launch angle for both balls has deviated exactly the same amount from this optimum, they will travel the same distance. Answer B is thus correct.

FORCE

THINK OF FORCE AS: Force = any influence capable of causing a mass to accelerate. o Examples of Force: The force due to gravity, contact forces, electrostatic forces, torque (a force at a distance from a point of rotation), tension, magnetic forces, etc. o Forces are vectors and can thus sum to zero. This is called "no net force" and is the same as if no force existed at all. If one of those forces is suddenly changed, a net force is instantly created. o Force is measured in Newtons.

Newton's First Law: The Law of Inertia.

THINK OF INTERTIA AS: The ability of an object to resist a change to its velocity THINK OF MASS AS: Mass = a measure of an object's inertia Centers of Mass, Gravity and Buoyancy: Center of Mass = a weighted average of mass distribution. To calculate the center of mass of multiple objects use the following formula, where r is the displacement vector between a reference point and each mass: C mass = (r1m1 + r2m2 + r3m3 . . . )/m total. Hint : First choose your reference point, the "origin of coordinates, "from which to measure each displacement vector. If you choose the center of mass of one object as the origin, one of the terms will drop out of the equation. Center of Gravity = at the center of mass Center of Buoyancy = at the center of mass of the fluid displaced by the submerged object (NOT at the center of mass of the submerged object itself).

Q7. Track the velocity of a ball as it is shot from a cannon and then returns to the ground (include air resistance).

Taking air resistance into account, the cannonball experiences a force due to the gases from the explosion inside the cannon that pushes it down the barrel of the cannon. The cannonball simultaneously experiences a force due to friction that opposes its motion. There is obviously a large net force oriented toward the end of the barrel when these are summed together. Once the cannonball leaves the barrel it becomes a projectile. Unlike the previous example, air resistance will create an opposing force on the cannonball throughout its flight and therefore a negative acceleration. This will occur in both the vertical and horizontal directions. The cannonball will not go as far, as high, or be in the air as long as it would have been without air resistance. When it strikes the ground it will have a horizontal velocity less than it did when leaving the barrel. The vertical velocity will also be less than the original vertical velocity because projectile motion is only symmetrical without air resistance. With air resistance, projectiles fall more steeply than their launch angles, decreasing total velocity at impact compared to initial total velocity. (We'll assume the end of the barrel and the ground are at the same height, though they logically would not be. If the ground is lower than the end of the barrel, the final velocity will be slightly higher than otherwise.)

What is the buoyant force always exactly equal to?

The buoyant force is always exactly equal to the weight of the amount of fluid displaced by the object.

IMPORTANT NOTE: F = Gmm/r2

The formula F = Gmm/r2 gives the force due to gravity NOT gravity itself. Gravity itself, usually called "gravity," "the strengthofthegravitationalfield,"or "accelerationduetogravity"isrepresented bylowercasegandisdescribedbythe formula: g = Gm/r2. Many students mistakenly reference the first formula when asked about gravity.

Q35. Why does V = √(2gh) work for either falling bodies or a mass on an inclined plane?

The formula V = √2gh is derived from conservation of energy by equating mgh to 1/2mv2 and solving for v. As long as friction, air resistance, etc. are ignored, energy will be conserved in an identical way whether the object falls directly to the ground or rolls down a plane.

Q3. A rock is thrown into the air at an angle of 30° from vertical at a velocity of 40 m/s. Resolve this vector into its vertical and horizontal components.

The mistake most students will make on this easy problem is to use the wrong formula for each component. Note that if measured w.r.t. the x-axis then the x-component is given by Vcosθ; but if the angle is measured w.r.t. the y-axis then the x-component is given by Vsinθ. We suggest it is easiest and the least error-prone to make it a habit to always convert angles so that they are measured w.r.t. the x-axis, then always use Vcosθ for the x-component and Vsinθ for the y- component. For this question, the x-component is given by: Vcosθ = (40m/s)(cos60°) = (40)(.5) = 20m/s. The y-component is given by: Vsinθ = (40m/s)(sin60°) = (40)(.9) = 36m/s.

Q30. Why must we include a negative in the above equation? (Hint: Without the negative sign, what does this equation predict for increasing r between two planets?)

The negative sign is necessary because without it the formula would predict that as r increases PE decreases. If you have an even basic concept of PE you will see that cannot be. A rock gets more PE as it gets farther from the center of the earth, not less. The negative makes it so that as r increases we get a smaller negative number, which is actually a larger value.

Linear Motion Displacement:

The shortest distance between point A and point B.

Q16. What does the sign of the slope on a velocity vs. time graph tell you? Does this tell you that the particle is moving to the right or to the left?

The sign of the slope on a velocity vs. time graph tells us the sign of acceleration. This tells us which direction the acceleration vector points and hence the direction of the net force, but it tells us absolutely nothing about which way the particle is moving. Remember that velocity and acceleration can be—and often are—oriented in opposite directions.

Q4. Determine the units included in a Newton using the equation F = ma.

The units of a Newton can be found from F = ma. The SI unit for mass is kg and the SI unit for acceleration is m/s2. Therefore, the units of a Newton are: kg*m/s2.

What causes the buoyant force?

There are a few ways to conceptualize the cause of the buoyant force. First, it is a characteristic of liquids (as compared to solids) that liquids can permanently resist a compressive force applied normal to their surface, but not parallel to their surface. Solids permanently resist both (up to the elastic limit of the solid). This gives us a rather general sense for the fact that liquids want to push objects up toward their surfaces. However, we think the best way to intuit buoyant force is to look at the pressure differential between the top and bottom of an object.The fluid pressure, pgh, will be larger at the object's bottom surface than it is at the top surface due to the larger value of the h term for the deeper surface. Let's examine a submerged cube with the same surface area both top and bottom. The formula P = F/A tells us that if pressure is greater at the bottom and area stays the same, there must be a greater force up on the bottom surface than there is down on the top surface. This makes it logical that any submerged object will experience a net upward force due to pressure differential.

Q23. If a man on a cliff first fires a gun straight down toward the ground below; and then fires a second round straight up into the air, which bullet will hit the ground with the greater velocity? (Assume both bullets hit the ground at the base of the cliff; neglect air resistance.)

They will both have the same final velocity. The muzzle velocity of the gun will be the initial velocity for the case where the gun is pointed straight up. Because projectile motion is symmetrical, the bullet will return to exactly this same muzzle velocity as it passes the top of the cliff from which it was fired. This will be equivalent, therefore, to turning the gun downward and firing it straight at the ground from the top of the cliff.

Q28. Imagine two planets of masses A and B, where A = 2B. What is the ratio of the forces between them? What is the ratio of the acceleration between them?

This is an excellent tutorial on how the MCAT tests. In our experience most MCAT prep books and companies lead students to believe that the "hardest" MCAT questions are those that require the most obscure or advanced knowledge. To the contrary, we almost never see an MCAT question that ventures outside of what we consider to be a fair, reasonable, moderate conceptual understanding of topics that would be EMPHASIZED in almost any sophomore version of these prerequisite classes. How the MCAT generally makes the hardest questions "hard" is by testing simple things that require very careful, precise thinking or reasoning. For example, this question tests your understanding of action-reaction forces and acceleration with regard to planets. That's pretty basic. However, what makes this question challenging is that you must carefully interpret the ratio given correctly, obtain the correct information, apply it correctly, and produce the correct answer without being tempted into one of various possible misconceptions or errors. Put another way, we often see two students who have nearly identical levels of mastery of the science. However, one will score many points higher than the other because that student is better at engaging in careful, logical thought without making errors. Yes, this was a long aside, but helping your students truly grasp the reality of how they are being tested on the MCAT, and helping them alter their study approaches as a result, will have a far bigger influence on their score performance than will this single question. Now, as for the question itself. Because A is 2B, A is the larger planet by a factor of 2. The ratio of the forces between them is still 1:1 according to Newton's Third Law. However, under the same force, the smaller planet will have twice the acceleration. This means the ratio of their accelerations would be 1:2.

Apparent Weight:

This is an important point. The apparent weight of a submerged object is the actual weight minus the buoyant force.

Q13. Distinguish between the following: velocity and speed.

Velocity is the rate of change of displacement. Speed is the rate of change of distance. They will NOT be equal in most situations. Once again, the racecar driver may have very high speed but has an average velocity of zero every time he passes the starting line.

Q14. Acceleration is any change in _____. Do you accelerate when you walk around a corner at constant speed?

Velocity. Yes, you accelerate when you walk around a corner because velocity is a vector with both magnitude and direction and a change to either magnitude or direction constitutes a change—and therefore acceleration.

Weight

Weight is the force acting on an object due to gravity. Assuming you don't eat, sweat, cut your hair, shave, or use the restroom, your mass will not change regardless of what planet you're on. Your weight, however, could change from location to location or planet to planet.

Linear Motion Graphs:

You will mess these up unless you follow the steps outlined below each and every time. Even then, most students are likely to slip up a time or two before they get it down. Seriously, just follow the steps outlined below. When students do not follow these steps, analyzing and double-checking the meaning of what they see graphically and mathematically, they always come up with answers that seem so logical to them based on what the graph "looks like" but are nevertheless completely wrong. Steps for Interpreting Linear Motion Graphs. Ask yourself the following: 1) What does the slope represent? For example, on a displacement vs. time graph the slope represents velocity and on a velocity vs. time graph it represents acceleration. Be careful, however. It is the actual numerical value of the slope of the line that represents velocity or acceleration in each case, NOT how the line "looks." A straight line with a positive slope "looks" like it is "increasing" at first glance. However, if the line is straight the slope is NOT increasing, the slope is constant. It is the value on the y-axis that is actually increasing. SLOPE = VELOCITY "As the slope goes, so goes the velocity." Whatever characteristics the velocity has, the slope will exhibit the same (and vice versa). If the velocity is constant, then the slope is constant (i.e., a straight line). If the velocity is changing, then the slope is changing (i.e., a curved line). If the velocity is positive, then the slope is positive (i.e., moving upwards and to the right). This very principle can be extended to any motion conceivable. 2) Is this slope (+) or (-)? What does the sign of the slope tell you? For example, a positive slope on a displacement vs. time graph tells us that velocity is positive and therefore by convention the motion is to the right. 3) Is the slope constant (straight line) or non-constant (curved line)? What does this observation tell you? For example, a straight line on a displacement vs. time graph tells us that velocity is constant and therefore the object is in equilibrium. 4) What value is on the y-axis? Students tend to forget this important information. It is the quantity on the y-axis that is changing with time. If we see a flat line with zero slope this means that whatever is on the y-axis is constant. 5) Is the y value (+) or (-) (i.e., are you above or below the x-axis?). What does this observation tell you? For example, on a displacement vs. time graph when we are above the x-axis displacement is positive and we are to the right of the origin. When we are below the x - axis displacement is negative and we must be to the left of the origin. On a velocity vs. time graph being above or below the x-axis tells you the sign of the velocity: whether the object is moving to the right or to the left. 6) Do you expect the value on the y-axis to be large or small at t = 0? For example, for a ball being thrown up into the air many students will pick an erroneous velocity vs. time graph that features a line starting at the origin. That's impossible. If initial velocity is zero the ball won't travel upward! Velocity should be a high value at the start and decrease with time. This simple step very often rules out 2 or 3 possible answers choices on this question type.

Q2. True or False? a) The product of two vectors is always a vector. b) The product of a vector and a scalar is always a vector.

a) False. The product of two vectors can equal a scalar. An example would be Work = Fdcosθ. It can also equal a third vector, as in T = Frsinθ. b) True. A vector multiplied by a scalar is always a vector with its magnitude increased by the factor of the scalar by which it was multiplied (or decreased, if the scalar is less than 1).

Q1. Label the following as vectors or scalars: mass, temperature, velocity, speed, displacement, acceleration, force, work, energy, weight, charge, electric field, magnetic field, time, momentum, impulse, density and torque.

mass = scalar; temperature = scalar; velocity = vector; speed = scalar; displacement = vector; acceleration = vector; force = vector; work = scalar; energy = scalar; weight = vector; charge = scalar; electric field = vector; magnetic field = vector; time = scalar; momentum = vector; impulse = vector; density = scalar; torque = vector.

Newton's Second Law:

o F net = ma, where F net is the vector sum of all forces acting on the object o All objects follow F net = ma. The MCAT will try to make problems sound difficult that can really be easily solved using Newton's Second Law. Whenever force, mass, or acceleration are asked for, first try to solve using Fnet = ma. o Students tend to have two common misconceptions regarding force and acceleration. Make sure you understand the following: First, a constant force will NOT cause an object to accelerate faster and faster; it will cause a constant (non-changing) acceleration. Only a changing force can cause a changing acceleration. That being said, a constant force can cause an object to move faster and faster—in which case the displacement is changing non-linearly, the velocity is changing linearly, and the acceleration is not changing at all (i.e., it is constant). A related misconception is that a constant force applied to an object will cause it to travel at a constant velocity. This is absolutely NOT true. A constant net force will always cause a constant acceleration—and therefore a CHANGING velocity. Rehash these relationships until they are perfectly clear in your mind. Second, you CANNOT accelerate a ball horizontally across the room by throwing it. To accelerate, an object must be either 1) in contact with the object creating the force, or 2) be under the influence of a field force (e.g., gravitational or electrical) at that exact moment. Thus, the ball only accelerates horizontally during the brief time it is in contact with the object creating the horizontal force—in this case, your hand. In the vertical direction the ball is always accelerating (after it leaves your hand) because it is always under the influence of earth's gravitational field.

Gravitational Potential Energy:

o PE = mgh (near earth) Anything with mass can have gravitational potential energy. For example, fluids have mass, so they can also have potential energy. However, because they don't always move as a single unit, it is more useful to replace the mass term in mgh with density ρ (mass/volume) to give PE per unit volume of fluid = ρgh. o PE = -Gm1m2/r (in space, or near earth if NOT assuming g = 10m/s2)

Q17. For a) a velocity vs. time graph, and b) a displacement vs. time graph, describe what happens to a particle's motion under each scenario: 1) the line crosses the x-axis, 2) the line forms a sharp corner, and 3) the line is perfectly horizontal.

o Velocity vs. Time: 1) Crossing the x-axis means velocity went from positive to negative; in other words, the particle turned around.; 2) A corner (where slope abruptly changes from positive to negative or vice versa) tells us that the direction of the acceleration vector abruptly reversed; 3) A horizontal line tells us that acceleration is zero (because slope is zero) and that velocity is constant. o Displacement vs. Time: 1) Crossing the x-axis means that we passed the origin; 2) A corner tells us that the sign of velocity (which is the slope of a d vs. t graph) abruptly changed— in other words, the object turned around [NOTE to your students that this is entirely different from a velocity vs. time graph where the action of "turning around" is indicated by crossing the x-axis]; 3) A flat line tells us that displacement is constant, which means the object is standing still. It also tells us that velocity is zero because the slope of the line is zero—another proof that the object is not in motion at that instant.