## Wednesday, February 1, 2017

### Acceleration and Average Velocity

When an object changes position, the rate of that change is velocity. When an object changes velocity, the rate of that change is acceleration.

To find how far an object moves—that is to say, to find the magnitude of an object's displacement—the rate of position change is multiplied by the amount of time that it changes. Or, as an equation,

d = vt

where d is change in position (displacement), v is rate of change in position, (average velocity), and t is elapsed time.

In some cases, displacement is not measured from the point at which motion begins, but is measured from some other fixed point. In those cases, the total displacement becomes the sum of the initial displacement (initial position) and the distance moved. In such a case,

df = di + vt

where df is the final displacement from the fixed point, di is the initial displacement from the fixed point, v is rate of change in position, (average velocity), and t is elapsed time.

What is true for displacement closely parallels what is true for acceleration.

The change in velocity can be found by multiplying the rate that velocity changes (acceleration rate) by the amount of time that it changes (elapsed time). As an equation,

v = at

where v is the change in velocity, a is the rate of acceleration, and t is the elapsed time.

In many cases, an object is already in motion (has an initial velocity) when the acceleration begins. Logic and reason leads to the conclusion that the final velocity of an object is the initial velocity plus or minus the change in velocity. Since change in velocity is found (see above) easily, the equation for final velocity is

Vf = Vi + at

where Vf is final velocity, Vi is initial velocity, a is the rate of acceleration, and t is the elapsed time.

In many cases, however, Vi is 0 (zero), so the formula simplifies to the change in velocity formula.

To extend and begin to combine the above principles of motion, it is necessary to first consider how average velocity might be found given acceleration and time. Fortunately, this is relatively logical and easy to reason out. And the math on it is equally easy!

To apply logic, if something accelerates at a constant rate, it starts our at one velocity and moves faster and faster until it reaches a final velocity. Suppose it starts out a 0 m/s and ends up at 4 m/s. Through the course of its acceleration, it passes through all of the velocities between 0 and 4 m/s.

That might look like this:

0 sec, v = 0
2 sec, v = 1
4 sec, v = 2
6 sec, v = 3
8 sec, v = 4

The data above shows and elapsed time of 8 seconds, and in that time, the velocity of the object changed from 0 to 4 m/s. It might be tempting to just guess that, since the final velocity was 4 m/s, over the 8 seconds, the average velocity was 2 m/s. If that guess was made, the answer would be correct.

The list below gives Vf, Vi, and Vave:

Vf= 0, Vi=6, Vave= 3
Vf= 10, Vi=16,, Vave= 13
Vf= 0, Vi=10, Vave= 5
Vf= 100, Vi=160, Vave= 130
Vf= 4, Vi=8, Vave= 6

Simply allowing logic and reason to stand alone would violate the need of science to describe everything with equations! So, there is an equation:

Vave = (V + Vi) / 2

So what if Vi is zero? The math gets even easier!

The ability to find an average velocity allows more work with acceleration to be done. Since

if the average velocity is known, the displacement can be found. The work can be done in three steps.
• First, find the final velocity.
• Next, find the average velocity.
• Last, use the average velocity and the elapsed time to find the displacement.

EXAMPLE:

A car with an initial velocity of 20 meters per second (m/s) accelerates at a uniform rate of 3 (m/s)/s for 10 seconds. How far does it travel while it is accelerating?

Step 1.) Find the final velocity.

vf = vi + at
vf = 20 + 3 • 10
vf = 20 + 30
vf = 50 mph

Step 2.) Find the average velocity.

v(ave) = (vf - vi)/2
v(ave = (50 + 20)/2
v(ave) = 70/2
v(ave) = 35 mph

Step 3.) Find the displacement.  NOTE: the elapsed time is the same for displacement and acceleration.

df = di + vt
df = 0 + 35 • 10
df = 350 meters (350 m)

By combining an understanding of average velocity with the ability to find displacement and to find final velocity, it becomes possible to calculate the displacement that occurs during a period of acceleration.  This 3-step method should be used when given acceleration and time and asked to find distance.