Physics Ch 3 Test
What is the time For Object Dropped Vertically vs. Object Projected Horizontally?
1. **Galileo predicted that an object projected horizontally will reach the ground in the same time an an object dropped vertically; 2. this is because the vertical motions are the same in both cases, and thus, the vertical position of the ball is the same
What is the Graphical Notation of Vector Angles and Components?
1. **NOTE: By convention, θ is chosen to be the angle that the vector makes with the positive x axis; 2. thus, in order to find angle θ of resultant, arctan is always: tan-1(vy/vx):
What is a Scalar Quantity?
1. A quantity that has no direction associated with it; 2. specified completely by a number and units 3. (ex: mass, time, and temp)
What is a Vector Quantity?
1. A quantity with both direction and magnitude; 2. (number and sign); 3. Ex: velocity not only refers to how fast something is moving, but also its direction 4. (other vector quantities are displacement, force, and momentum)
How do we identify quadrants in the component method?
1. Always be attentive about the quadrant in which the resultant vector lies; 2. the signs of trig functions depend on which "quadrant" the angle falls in, and the best way to keep track of angles and to check any vector result is to draw a diagram)
What are the Separate x and y Conventions for Projectile Motion Problems?
1. As we treat horizontal and vertical components of velocity (vx and vy) separately, we can apply the constant acceleration equations to the x and y components of motion separately; where: 2. X and y are the respective displacements, vx and vy are the component so velocity, ax and ay are the components of acceleration (each of which is constant); 3. subscript 0 means t=0
How is graphical addition imprecise?
1. Because adding vectors graphically with a ruler and protractor is often not sufficiently accurate and not useful for vectors in three dimensions, 2. a more precise method for adding vectors involves breaking them into components/right triangles
What is the Parallelogram Method of Graphically Adding Vectors?
1. Equivalent to tail-to-tip method, in which two vectors are drawn starting from a common origin, and 2. parallelogram is constructed using these two vectors as adjacent sides, in which resultant is the diagonal drawn from the common origin:
What is an Example Involving Range of Vector Lengths?
1. Ex: If two vectors each have length 3.0 units, what is the range of possible lengths for the vector representing the sum of the two? 2. Solution: The sum can take any value from 6.0 (=3.0+3.0) where the vectors point in the same direction, to zero (3.0 - 3.0) when the vectors are antiparallel
What is an Example Involving Moving Wagon and Horizontal and Vertical Velocity?
1. Ex: Someone sits upright in a wagon, which is moving to the right at constant speed. The child extends her hand and throws an apple straight upward, while the wagon continues to travel forward at constant speed. If air resistance is neglected, will the apple land behind the wagon or in front of the wagon? 2. Solution: While she throws apple straight up from her own reference frame, when viewed by someone on the ground, the apple will follow path of a projectile (Experiences no horizontal acceleration, so vx0 will be constant and equal to speed of wagon, and as the apple follows its arc, the wagon will be directly under the apple at all times because they have same horizontal velocity; thus, when the apple comes down, it will drop into her hand (b) **NOTE: If air resistance were accounted for, ball would land behind wagon
What is a Conceptual Example Involving Crossing a River?
1. Ex: man in motor boat is trying to cross river that flows due west with a strong current; the man starts on the south bank and is trying to reach the north bank directly from his starting point. Should he travel/head due north, head due west, head in northwesterly direction, or head in northeasterly direction? 2. Solution: If man heads straight across river, current will drag boat downstream (west); to overcome river's westward current, boat must acquire eastward component of velocity and northward component; thus boat must travel in northeasterly direction
What are the Constant Acceleration Equation Conventions for Y Component of Projectile Motion (Vertical Component)?
1. For y component of motion, at instant ball leaves table top (t=0), it has only an "x" component of velocity; once ball leaves table (t=0), it has vertically downward acceleration "g" (gravity) and thus vy is initially zero (vy0 = 0), increasing continually until ball hits the ground; this applies only to non-upward angled projectile motion (when object starts horizontally) 2. If y is positive upward, then ay = -g; thus from equation v=v0+at, can write vy = -gt for vertical component of motion because vy0 = 0; 3. The vertical displacement of motion, derived from Δx = v0t + ½ at, can be written as Δy = -½ gt2, as ay = -g and v0y = 0
What are the Components of a Vector?
1. If a vector lies in a particular plane, it can be expressed as the sum of the two other vectors, called the components of the original vector; 2. the process of finding the components is known as "resolving the vector into its components"
What are the rules for For Objects Projected Upwards in projectile motion problems?
1. If an object is projected at an upward angle, the analysis is similar, except now there is an initial vertical component of velocity, yy0 2. (ex: as in finding horizontal muzzle velocity vs. initial velocity of ball when muzzle is angled upward); 3. Because of downward accel. Of gravity, vy usually gradually decreases with time until object reaches highest point on path, at which point (vy = 0, and thus, min. Speed = vx at top of trajectory), and then object moves downward and vy increases in downward direction (equal in mag, opp in sign-min velocity of object)
How can we find vector components given angle and hypotenuse?
1. If the sine, cosine, and tangent are as given in the figure, if we multiply sinθ = vy/v (where v = hyp and vy = y component) by "v" on both sides, we get: vy = vsinθ; 2. Similarly, if multiply cosθ = vx/v (where vx = x component) by v, we get: vx = vcosθ
What is a Conceptual Example Involving Balls Thrown at Different Angles?
1. If two cannonballs, one with larger angle, but both fired with identical initial speeds, 2. (a) Cannonball A, with the larger angle, will reach a higher elevation. It has a larger initial vertical velocity, and so by Eq. 2-11c, will rise higher before the vertical component of velocity is 0. 2. (b) Cannonball A, with the larger angle, will stay in the air longer. It has a larger initial vertical velocity, and so takes more time to decelerate to 0 and start to fall. 3. (c) The cannonball with a launch angle closest to 45o will travel the farthest. The range is a maximum for a launch angle of 45o , and decreases for angles either larger or smaller than 45 o
What are vector diagrams for relative velocities?
1. In determining relative velocity, it is important to construct a diagram; 2. in which each velocity is labeled by two subscripts: the first refers to the object, the second to the reference frame in which it has this velocity
How do trigonometric functions relate to the component method of resolving vectors?
1. In order to add vectors using component method, must use trig functions sine, cosine, and tangent 2. (given any angle θ, right triangle can be constructed by drawing line perpendicular to either of its sides, with h, o, and a representing hyp, opp, and adj sides respectively 3. (ratio of lengths of sides do not depend on the size of the angle):
What is the Step-by-Step Procedure for Problem-Solving With Projectile Motion?
1. Read each problem carefully and choose the objects you are going to analyze 2. Draw careful diagram showing what is happening to the object 3. Choose an origin and an xy coordinate system 4. Decide on a time interval (for projectile motion, can only include motion under effect of gravity alone, not throwing or landing); time interval must be same for x and y analyses (x and y motions are connected by a common time)
How do we add vectors along the same line?
1. Simple Arithmetic can be used for adding vectors if they're in the same direction/along the same line 2. (ex: if someone walks 8 km east one day and 6km the next, they'll be 8 km + 6 km = 14 km east of point of origin; for subtraction, if person walks east 8 km and then 6 km west-reverse direction-then displacement is obtained by subtraction: 8 km - 6 km - 2 km):
How do we know When Subscript Equations Involving Relative Velocity are Correct?
1. The equations involving relative velocity will be correct when adjacent inner subscripts are identical and when the outermost ones correspond exactly to the two on the velocity on the left of the equation 2. (this only works with plus signs-on the right-not minus signs)
What is Projectile Motion?
1. The general motion of objects moving through the air in two dimensions near the earth's surface 2. (ex: gold ball, thrown or batted baseball, etc.); 3. in examining projectile motion, we will ignore air resistance, and will be concerned with motion after the ball has been projected, and before it lands or is caught; 4. moving freely under the act of gravity alone
What is the Notation and Conventions for Projectile Motion?
1. We assume that the motion begins at time =0 and at the origin of the xy plane (so x0 = y0 = 0); 2. if analyzing a ball rolling off a table with an initial velocity in horizontal direction vx0, the vel. Vector at each instant points in direction of ball's motion at that instant and is always tangent to the path:
What are the Two Ways to Specify a Vector in a Coordinate System?
1. We can give its components Vx and Vy 2. We can give its magnitude V and the angle θ it makes with the positive x axis (where we use the pythagorean theorem for resultant magnitude and use arctan(vy/vx) for resultant angle θ)
How do we determine the Relative Velocity for Objects Not Traveling Along the Same Line?
1. When velocities are along the same line, simple addition or subtraction is sufficient to obtain relative velocity, 2. but if they're not along the same line, we must use vector addition-using the component method (when specifying a velocity, we must specify what the reference frame is)
How do we describe acceleration in terms of constant speed and velocity?
A particle with constant speed can be accelerating, if its direction is changing. Driving on a curved roadway at constant speed would be an example. However, a particle with constant velocity cannot be accelerating - its acceleration must be zero. It has both constant speed and constant direction.
What is the Relationship Between Displacement and Distance/Path of Object?
As length of path will always be greater than or equal to displacement, displacement will never be longer than path; will always be equal to or shorter than path
What is the Rule for Balls Thrown at Different Angles Thrown at Same Height?
For two balls thrown in the air at different angles, but reaching the same height, the one thrown at a shallow angle will reach ground in same time as one thrown at steeper angle, because gravity will act the same either way
What are Horizontal and Vertical Components of Projectile Motion?
Galileo showed that projectile motion could be understood by analyzing the horizontal and vertical components of motion separately:
What is a Conceptual Example Involving Time Traveled Across River By Two Rowers Contingent on Mode of Travel?
If two rowers, who can row at same speed in still water, cross river at same time, and one heads upstream at angle and the other heads straight and is pulled downstream, the one that head straight reaches opposite side first; Both rowers need to cover the same "cross river" distance. The rower with the greatest speed in the "cross river" direction will be the one that reaches the other side first. The current has no bearing on the problem because the current doesn't help either of the boats move across the river. Thus the rower heading straight across will reach the other side first. All of his rowing effort has gone into crossing the river. For the upstream rower, some of his rowing effort goes into battling the current, and so his "cross river" speed will be only a fraction of his rowing speed.
What is a Common Error in Parallelogram Method?
It is a common error to draw the sum vector as the diagonal running between the tips of the two vectors in the parallelogram method; however, this is incorrect, as it does not represent the sum of the two vectors
What is the Table of Separate "x" and "y" Conventions For Projectile Motion Problems?
The above equations can be simplified for the case of projectile motion, as we know: And that if θ is relative to + x axis, then vx0 = v0cosθ, and vy0 = v0sinθ
How do we extend the tail-to-tip method with three or more vectors?
The tail-to-tip method can be extended to three or more vectors; resultant is drawn from tail of first vector to tip of last;
What is a Diagram of Relative Velocity of Boat?
To move directly across the river, the boat must head upstream at an angle θ; velocity vectors are shown as green arrows
What are the three trig functions?
Where values of sin, cos, and tan don't depend on how big triangle is, only on size of angle θ;
What are the Kinematic Equations/Constant Acceleration Equations Tailored to Projectile Motion?
where y is positive upward, ax = 0, ay = -g; + becomes - due to "-g"