## Saturday, February 4, 2017

### Acceleration and Average Velocity—Finding Acceleration

Working with acceleration in relationship to elapsed time allows a great deal of information about an objects motion to be determined. The basic formula connects acceleration and time to final velocity.

vf = vi + at

were vf is final velocity, vi is initial velocity, a it the rate of acceleration, and t is elapsed time.

When combined with the distance formulas, acceleration and time can be used to find the total displacement. The three steps to do that are to:

1. Find the final velocity using vf = vi + at.
2. Find the average velocity using v(ave) = (vf + vi)/2
3. Use the average velocity to calculate displacement using df = di + vt

With a little manipulation, it is possible to go from use displacement and elapsed time to find the rate of acceleration. The process will be developed through this example:

An object moves a total of 10 meters (10 m) while it accelerates from rest for 5 seconds (5 s). What was the rate of acceleration?

To find the answer, the process is essentially to work through the steps that lead to finding the displacement backwardsIt is necessary to think about  first identify the given information.

df = 25 meters
di = 0
vf = unknown
vi = 0 (accelerates from rest)
a = that is what is being asked for
t = 5 seconds

Step 1.) Use the displacement information to find an average velocity (v).

df = di + vt
25 = 0 + v•5
25 = 5v
25/5 = v
5 m/s = v

Step 2.) Use the average velocity to find the final velocity (vf).

v(ave) = (vf + vi)/2
5 = (vf + 0)/2
5 = vf/2

Multiply both sides of the equation by 2 because, algebra.

2 • 5 = vf/2 • 2

10 m/s = vf

Step 3.) Use the final velocity to find the rate of acceleration (a).

vf = vi + at
10 = 0 + a • (5)
10 = 5a
10/5 = a
2 (m/s)/s = a

Thus, the rate of acceleration was 2 (m/s)/s.

While the math my look intimidating to to developing mathematicians, it is relatively simple algebra requiring only adding, multiplying, and dividing. This 3-step method needs to be used anytime the questions ask for acceleration and gives distance and time.