Mcat physics chapter 1 -Kinematics and Dynamics (kaplan)

speed

(v) is the rate of actual distance traveled in a given unit of time.

difference between base and derived units

Base units are the standard units around which the system itself is designed. Derived units, as the name implies, are created by associating base units with each other

What is the displacement of a man who walks 2 km east, then 2 km north, then 2 km west, and then 2 km south?

While his total distance traveled is 8 km, his displacement is a vector quantity that represents the change in position. In this case, his displacement is zero because the man ends up the same place he started, as shown below.

Basic units for length, weight, and time

foot (ft), the pound (lb), and the second (s),

representation of vectors

may be represented by arrows; the direction of the arrow indicates the direction of the vector. The length of the arrow is usually proportional to the magnitude of the vector quantity. Common notations for a vector quantity are either an arrow or boldface. For example, the straight-line path from here to there might be represented by a vector identified as A or A. The magnitude of the displacement between the two positions can be represented as | A|, |A|, or A. Scalar quantities are generally represented with italic type: the distance between two points could be represented by d. In this book (and all books of the Kaplan MCAT Review series), we will consistently use boldface to represent a vector quantity and italic to represent the magnitude of a vector or a scalar quantity.

vector addition

the combining of vector magnitudes and directions The sum or difference of two or more vectors is called the resultant of the vectors. One way to find the sum or resultant of two vectors A and B is to place the tail of B at the tip of A without changing either the length or the direction of either arrow. In this tip-to-tail method, the lengths of the arrows must be proportional to the magnitudes of the vectors. The vector sum A + B is the vector joining the tail of A to the tip of B and pointing toward the tip of B.

instantaneous speed of an object

will always be equal to the magnitude of the object's instantaneous velocity, which is a measure of the average velocity as the change in time (Δt) approaches zero: v = lim (∆t → 0) .∆x/∆t where v is the instantaneous velocity, Δx is the change in position, and Δt is the change in time. As a measure of speed, instantaneous speed is a scalar number. Average speed will not necessarily always be equal to the magnitude of the average velocity. This is because average velocity is the ratio of the displacement vector over the change in time (and is a vector), whereas average speed (which is scalar) is the ratio of the total distance traveled over the change in time. Average speed accounts for actual distance traveled, whereas average velocity does not: v=∆x/∆t where v is the average velocity, Δx is the change in position, and Δt is the change in time. Consider the example given earlier regarding the Earth's orbit. In one year, the Earth travels roughly 940 million kilometers, but its displacement is zero: d = 9.4 × 10 8 km x = 0 km The average speed is a measure of distance traveled in a given period of time; the average velocity is a measure of the displacement of an object over a given period of time. While the average speed of the Earth over a year is about 30 kilometers per second, its average velocity is again zero:

The magnitude of the resultant vector is simply the product of the magnitudes of the factor vectors and the sine of the angle between them. In this case, because one is oriented in the x-direction and the other is in the y-direction, the angle between them is 90°.

|A| × |B| × sin 90° = 3 N × 4 m × 1 = 12 N·m The magnitude is therefore 12 N·m. Now, to determine the direction of C, start by pointing your right thumb toward the left (negative x-direction). Your fingers will point toward the top of the page (positive y-direction). Your palm is therefore pointing into the page. Now, to determine the direction of D, start by pointing your right thumb toward the top of the page (positive y-direction). Your fingers will point toward the left (negative x-direction). Your palm is therefore pointing out of the page. Therefore, C is 12 N·m [⊗ (into the page)] and D is 12 N·m [⊙ (out of the page)].

What are the magnitudes and directions of the resultant vectors from the following cross products: C = A × B and D = B × A?

A: X = -3 N, Y = 0 B: X = 0, Y = +4 m

MUlTIPlYInG VeCTors by oTher VeCTors

In some circumstances, we want to be able to use two vector quantities to generate a third vector or a scalar by multiplication. To generate a scalar quantity like work, we multiply the magnitudes of the two vectors of interest (force and displacement) and the cosine of the angle between the two vectors. In vector calculus, this is called the dot product (A ∙ B): A ∙ B = |A| |B| cos θ In contrast, when generating a third vector like torque, we need to determine both its magnitude and direction. To do so, we multiply the magnitudes of the two vectors of interest (force and lever arm) and the sine of the angle between the two vectors. Once we have the magnitude, we use the right-hand rule to determine its direction. In vector calculus, this is called the cross product (A × B): A × B = |A| |B| sin θ The resultant of a cross product will always be perpendicular to the plane created by the two vectors. Because the MCAT is a two-dimensional test, this usually means that the vector of interest will be going into or out of the page (or screen). There are multiple versions of the right-hand rule that can be used to determine the direction of a cross product resultant vector. Shown in Figure 1.6 is one method considering a resultant C where C = A × B: 1. Start by pointing your thumb in the direction of vector A. 2. Extend your fingers in the direction of vector B. You may need to rotate your wrist to get the correct configuration of thumb and fingers. 3. Your palm establishes the plane between the two vectors. The direction your palm points is the direction of the resultant C. Note that you may have learned a version of the right-hand rule that is different from what is described here. For example, some students learn to point the right index finger in the direction of A and the right middle finger in the direction of B; when one holds the thumb perpendicular to these two fingers, it points in the direction of C. It makes no difference which version of the right-hand rule you use, as long as you are comfortable with it and are skilled in its proper use.

luminous intensity

Unit: candela Symbol: cd

length

Unit: meter Symbol: m

MUlTIPlYInG VeCTors by sCAlArs

When a vector is multiplied by a scalar, its magnitude will change. Its direction will be either parallel or antiparallel to its original direction. If a vector A is multiplied by the scalar value n, a new vector, B, is created such that B = nA. To find the magnitude of the new vector, B, simply multiply the magnitude of A by |n|, the absolute value of n. To determine the direction of the vector B, we must look at the sign on n. If n is a positive number, then B and A are in the same direction. However, if n is a negative number, then B and A point in opposite directions. For example, if vector A is multiplied by the scalar +3, then the new vector B is three times as long as A, and points in the same direction. If vector A is multiplied by the scalar -3, then B would still be three times as long as A but would now point in the opposite direction.

vector

are numbers that have magnitude and direction. Vector quantities include displacement, velocity, acceleration, and force. For example, the newton—a unit of force—is derived from kilograms, meters, and seconds: 1 N = 1kg.m/s^2

scalar

are numbers that have magnitude only and no direction. Scalar quantities include distance, speed, energy, pressure, and mass.

The difference between a vector and scalar quantity

can be quite pronounced when there is a nonlinear path involved. For example, in the course of a year, the Earth travels a distance of roughly 940 million kilometers.However, because this is a circular path, the displacement of the Earth in one year is zero kilometers. This difference between distance and displacement can be further illustrated with vector representations.

distanced trave;ed

considers the pathway taken and is a scalar quantity.

An object in motion may experience a change in its position in space, known as

displacement (x or d). This is a vector quantity and, as such, has both magnitude and direction. The displacement vector connects (in a straight line) the object's initial position and its final position. Understand that displacement does not account for the actual pathway taken between the initial and the final positions—only the net change in position from initial to final

force.n

is a vector quantity that is experienced as pushing or pulling on objects. Forces can exist between objects that are not even touching. While it is common for forces to be exerted by one object pushing on another, there are even more instances in which forces exist between objects nowhere near each other, such as gravity or electrostatic forces between point charges. The SI unit for force is the newton (N), kg ⋅ m which is equivalent to one

metric system

may be given in meters, kilograms, and seconds (MKS) or centimeters, grams, and seconds (CGS).

current

unit-ampere (coulomg/second) symbol-A

Temperature

unit-kelvin symbol-K

Mass (not weight)

unit-kilogram symbol-kg

Amount of substance

unit-mole symbol-mol

time

unit-second symbol-s

____is a vector. Its magnitude is measured as the rate of change of displacement in a given unit of time, and its SI units are meters per second. The direction of the velocity vector is necessarily the same as the direction of the displacement vector.

velocity