# One-Dimensional Kinematics

**Kinematics** is the science of studying the positions and motions of objects as a function of time. It is not concerned with the causes of motion. It deals with the relationship between **displacement (d), velocity (v), acceleration (a), and time (t)**.

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**vector**is a physical quantity with direction and magnitude (e.g., velocity or force). Displacement, velocity, and acceleration are vectors.

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**scalar**is a quantity that has only magnitude and no direction. Time and temperature are examples of scalars. Speed is also a scalar quantity. It is the measure of how much distance is traveled within a certain time period.**Speed**, or how fast something goes, is expressed as distance/time (d/t). It is illustrated by the following picture.

Do not confuse instantaneous speed with average speed! Average speed is the distance traveled (d) in a given amount of time (t). It is expressed as distance per time (d/t). Instantaneous speed is the speed something is going at a certain point in time.

**Velocity** is a vector quantity. It is the speed and direction of an object. Average velocity = total displacement . Note that displacement time is NOT the same as distance traveled. For example, if a car travels 1 mile east then 1 mile west, the total displacement is 0 because the car would end up back at the starting point! Thus, average velocity is 0. The total distance traveled would be 2 miles and the speed would be 2/t. Velocity and direction of acceleration are illustrated in the following illustration.

Try to determine the direction of acceleration and velocity above! As the car moves down the incline, it gains speed (accelerates). Along the straight sections of track, the car slows slightly due to air resistance. This again is acceleration (change in velocity). Around the curve, the car is changing direction which means it is accelerating. Remember, acceleration is related to a change in speed or direction. What is the direction of the acceleration vector? Follow the rule of thumb that the direction of the acceleration is that an object which is slowing down will have an acceleration directed in the direction opposite of it's motion.

In the animation below, which car(s) are accelerating? Which car experiences the greatest acceleration?

The acceleration of a body moving only under the force of gravity is g, the gravitational (free-fall) acceleration, which is directed vertically downward. On Earth, g = 9.8 m/s2 (32 ft/s2). On the Moon, acceleration due to gravity is 1.6 m/s2.

**Graphical interpretations for motion along a straight line (x-axis) are as follows:**

1. The instantaneous velocity of an object at a certain time is the slope of the displacement versus time graph at that time. It can be positive, negative, or zero.

2. The instantaneous acceleration of an object at a certain time is the slope of the velocity versus time graph at that time.

3. For constant-velocity motion, the x-versus-t graph is a straight line. For constant-acceleration motion, the v-versus-t graph is a straight line.

4. In general (one, two, or three dimensional motion), the slope of any moment of the distance-versus-time graph is the speed.

**Review Questions:**

1. Scott realizes he has overslept and might miss physics so he jumps in his car and accelerates from rest to a speed of 30m/s in 12 seconds with a constant acceleration. What is the distance Scott traveled in 12 seconds?

2. Suppose Scott reaches 30m/s by the time he reaches school. He hits his brakes and produces a constant deceleration of 3m/s2. How far will he go before he stops?

3. Carla and Helena are driving down a long, straight stretch of the freeway. Carla, who is in front, is going 15 m/s and continues to travel at constant velocity. Helena, who is behind her, is traveling at 30 m/s. When Helena is 200 meters behind Carla, she hits her brakes producing a constant deceleration of 0.25 m/s2. At what time after Helena hits her brakes do they collide?

4. Lorena is 1.8 meters tall. She gets upset over her new Korn CD and throws it straight up in the air at an initial velocity of 10 m/s. (a) How long does it take for the CD to hit her on the head? (b) If she stepped out of the way, how long would it take for the CD to hit the ground from the time it missed the top of her head?

5. Eugene is on the roof of a tall building (100 meters) and decides to drop a water balloon on Kali. He drops a water balloon weighing 2 lbs and a rock that weighs 10 lbs. Assumed there is no air resistance, when do these objects land on Kali?

Equations:

v = vo + at

d = vot + (1/2)at2

v2 = vo2 + 2ad