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 usingvf = vi + at.2. Find the average velocity usingv(ave) = (vf + vi)/23. Use the average velocity to calculate displacement usingdf = 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

*backwards*. It is necessary to think about first identify the given information.df = 25 metersdi = 0vf = unknownvi = 0 (accelerates from rest)a = that is what is being asked fort = 5 secondsStep 1.) Use the displacement information to find an average velocity (v).df = di + vt25 = 0 + v•525 = 5v25/5 = v5 m/s = vStep 2.) Use the average velocity to find the final velocity (vf).v(ave) = (vf + vi)/25 = (vf + 0)/25 = vf/2Multiply both sides of the equation by 2 because, algebra.2 • 5 = vf/2 • 210 m/s = vfStep 3.) Use the final velocity to find the rate of acceleration (a).vf = vi + at10 = 0 + a • (5)10 = 5a10/5 = a2 (m/s)/s = aThus, 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.

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