Distance (D) Is the Path Length Traveled by an Object (Measured in Meters Or Feet)

Chapter 3 Linear Motion

This chapter is concerned with one-dimensional motion. Since there are still choices as to direction, we must be careful in assigning quantities either positive or negative. In nature there are two types of quantities: scalars and vectors. Scalars require only a knowledge of magnitude (size) for a complete description. Mass or temperature are examples of scalars. Vectors include the concept of direction. Velocity and force are vector quantities. We generally use arrows to symbolize vectors. The length of the vector gives the magnitude and the orientation of the arrow gives the direction. Many times in discussing motion, we have scalar concepts which have a vector analogue. For example:

Distance (d) is the path length traveled by an object (measured in meters or feet).  Displacement ( d ) is the vector from the starting point to the ending point.  d B A d

Speed (measured in m/sec or ft/sec) is the scalar quantity that measures the rate at which distance is covered. Average speed is simply distance/time. Instantaneous speed, however, is how fast you are moving at an instant in time. It is the quantity measured by the speedometer in your car. The analogous vector quantity is velocity. Acceleration is the change in velocity and is measured in m/s2 or ft/s2. It is a vector and is used for any change in velocity – speeding up, slowing down, or turning). Whenever you observe that the velocity of an object is not constant, an acceleration is acting. The problems we will consider always have a constant acceleration.

Equations describing linear motion

Define d as the distance traveled, do the initial position, v the velocity at some point in the motion, vo the initial velocity, and a the acceleration. Two equations predict the position (distance traveled) and velocity at a time t after the start of the motion.

2 d = do + vot + ½at v = vo + at In many cases we are at liberty to start measuring at zero (do = 0) and the motion begins at rest (vo = 0), in which case these equations become

d = ½at2 v = at , but these are not new equations, merely adaptations of the general forms.

Free Fall

A special case of linear motion is free fall. The conditions are:

1. The only force at work is gravity – no atmosphere. 2. We are near the surface of the earth (so that the acceleration can be treated as constant)

The acceleration here is called g, the acceleration due to gravity at the Earth’s surface. It has a value of 9.8 m/s2 or 32.1 ft/s2. In this course we can approximate them this as 10 m/s2 and 32 ft/s2. Starting from rest at the origin, the equations above look like

d = ½gt2 = 5t2 (in metric units) v = gt = 10t (also in metric units)

Suppose we throw an object in the air. It starts with a velocity directed upward and we decide to call up positive. We set up the distance measurements to begin where the motion begins. The graphs of this situation look like

) 6 m ( 5 t n

c 2 a l

p 1 s i

D 0 0 0.5 1 1.5 2 Time (s)

) 10 s / 5 m (

y 0 t i

c -5 o l

e -10 V -15 0 0.5 1 1.5 2 Time (s) )

s 0 / s

/ -2 m

o -6 i t

a -8 r e l -10 e c

c -12 A 0 0.5 1 1.5 2 Time (s)

Notice: 1. Maximum height is reached (t = 1 sec) when the velocity is zero. The object comes momentarily to rest. 2. It takes just as long to rise as it takes to fall. 3. The speed at two seconds (back at the ground) is the same as the speed at the beginning. 4. The acceleration is constant and negative (we chose positive up). 5. The mass of the object doesn’t matter. All objects near the surface of the Earth fall at the same rate. Recall Galileo’s experiment dropping objects of vastly different mass.