SAT Physics Kinematics in One Dimension - Kinematic Quantities

SAT Physics Kinematics in One Dimension - Kinematic Quantities

KINEMATIC QUANTITIES
Kinematics involves the mathematical relationship among key quantities describing the motion of an object. These quantities include displacement—x, velocity—v, and acceleration—a. You should also note the relationships between displacement and distance and between velocity and speed.

Displacement and Distance
Displacement, x, is a vector extending from the initial position of an object to its final position. The variable x is typically used for horizontal motion while y and h (height) are used for vertical motion.
Displacement differs slightly from distance, d. Distance is a scalar quantity representing the actual path followed by the object. When an object travels in a straight line and does not reverse its direction, then distance and the magnitude of displacement are interchangeable.

Distinguishing Between Distance and Displacement

A person runs 40 meters in the positive x-direction and then 30 meters in the positive y-direction. Determine both the distance traveled by the runner and the magnitude of the displacement.
 
WHAT'S THE TRICK?

Distance is a scalar using the actual path followed.

d = 40 m + 30 m = 70 m

Displacement is a vector extending from the initial position to the final position. The vector drawn from the initial point to the final point becomes the hypotenuse of a 3-4-5 right triangle.

The resulting displacement is 50 m. The variable used to represent this displacement can vary. Although the 50 m displacement does not lie along the x-axis, some authors will report this displacement as x = 50 m or Ax = 50 m. Others may choose a letter that does not rely on an axis, such as d = 50 m. However, used this way can be confused with distance. You may encounter a variety of conventions, and you need to be flexible. Regardless of the variable chosen, focus on what is being solved (distance or displacement).

Velocity and Speed
Velocity and speed are kinematic quantities measuring the rate of change in displacement and distance. A rate is a mathematical relationship showing how one variable changes compared with another. When the word “rate” appears in a problem, simply divide the quantity mentioned by time. Velocity, v, is a vector describing the rate of displacement, Ax. The equation for velocity

solves for the average velocity during a time interval, t. Additional information is needed to determine if velocity is constant or is changing during the time interval.
If velocity is changing, then it has different values at different moments in time. However, instantaneous velocity is the velocity at a specific time, t. If you report that you are driving north at 65 mph, you have given an instantaneous velocity. This is a snapshot, freezing the problem at a specific instant. In kinematics, you will encounter two specific instantaneous velocities. Initial velocity, v_i, is at the start of a problem. Final velocity, v_f is at the end of a problem.
Velocity is a vector quantity, so it includes a specific magnitude and a direction. When the magnitude and direction of velocity are both constant, we say that the object is moving at constant velocity. However, when direction is changing, the tenn speed may be used. Speed is a scalar quantity that calculates the rate of distance, as opposed to displacement. If an object travels in a straight line, then the terms speed and velocity are interchangeable.

Distinguishing Between Speed and Velocity

Consider a car moving from point A to point B and then returning to point A.

This motion takes place in 20 seconds. Determine the speed and average velocity during the entire motion.
 
WHAT"S THE TRICK?

The object reversed direction and returned to its starting point. It traveled a specific round-trip distance. However, at the end of the motion, there was no overall displacement. Speed is a scalar that depends on actual distance moved by the object.

Average velocity is the rate of the displacement (straight-line distance from starting point to ending point)

Acceleration
Acceleration is the rate of change in velocity.

Table 3.2 The Effect of Acceleration on Velocity

When answering conceptual questions, the possibility of changing direction is often overlooked. A car moving around a circular track at a constant speed is said to have uniform acceleration as opposed to constant acceleration. The phrase “uniform acceleration” indicates that the magnitude of acceleration will remain the same while the direction may be changing. The phrase “constant acceleration,” however, indicates that both magnitude and direction are the same.

Conceptual Problem

Can an object be accelerating if it has constant speed?
 
WHAT'S THE TRICK?

Speed is a scalar. Constant speed implies only constant magnitude; it gives no information about direction. An object at constant speed may be changing direction. For example, a car moving around a circular track at a constant speed must continually change direction. Changing direction involves a change in the velocity vector, and a change in velocity during a time interval is acceleration. Therefore, an object with a constant speed may be accelerating. Had the problem specified constant velocity, the answer would be quite different. An object undergoing constant velocity must have both constant magnitude (constant speed) and constant direction, resulting in no acceleration.

Acceleration of Gravity
All objects on Earth are subject to the acceleration of gravity. This acceleration has a known value at Earth’s surface. It is so prevalent in physics problems that it receives its own variable, g. The acceleration of gravity acts downward and has a value of 9.8 m/s². For the SAT Subject Test in Physics, the value is rounded to 10 m/s².

IDENTIFYING VARIABLES
Correctly identifying the variables in a problem statement will assist you in the problem­ solving process. When presented with a word problem, begin by writing down all of the variables given in the question. Typically, you will find that you are given three variables and you must solve for a fourth. However, only one or two numerical values may appear in the word problem. In these situations, you may have overlooked variables hidden in the language of the problem.

Hidden Variables
Throughout physics, hidden variables appear in either of two main forms:

1. Known constants, such as gravity, do not need to be specifically mentioned in the text of a problem even though they are important in solving the problem.
2. When quantities have a zero value, they are often indicated by key phrases in the text of the problem.

The common hidden zero quantities used in kinematics are listed in Table 3.3. The wording used in the question will help you find these variables.

Table 3.3 Common Quantities with a Zero Value

Sign Conventions
The kinematic quantities—displacement, velocity, and acceleration—are all vectors. The magnitude of a vector is always positive. However, vectors can point in either a positive or a negative direction. When vectors point in a negative direction, a negative sign is added to the magnitude of the vector for calculation purposes.
The coordinate axis system, discussed in Chapter 2, is the best tool to use when determining the correct sign on vector quantities. Picture the object at the origin of the coordinate axes at the start of the problem {x_i = 0 and y- = 0). The default positive directions are right and upward. Any vector quantities pointing to the left or downward will include a negative sign in calculations. Table 3.4 summarizes the sign conventions for kinematic variables.

Table 3.4 Signs Used for Kinematic Variables

Identifying Variables

A ball is thrown upward with an initial velocity of 15 meters per second. Write a complete list of kinematic variables for the instant the ball reaches its maximum height.
 
WHAT'S THE TRICK?

Only one numerical value is mentioned. However, there are two hidden values in the word problem.

1. Unless told otherwise, all problems take place on Earth (g = -10 m/s²).
2. The phrase “maximum height” implies the ball has an instantaneous vertical velocity of zero (v_f = 0 m/s).

Adding these hidden variable results in the following variable list: