Example of Motion with Non-constant Acceleration

The following example shows the fallibility of the formula (U_A_M 2) when applied to motion with non constant acceleration.

An UFO moves at 20 m/s along a straight line for 700 m; then, suddenly it starts moving at 25 m/s along the same straight line for 600 m.

Find the average velocity of the UFO for the entire trip length.



 21.5 m/s



 22.0 m/s



 23.0 m/s



 22.5 m/s



None of the above.


In order to properly solve this problem, it is necessary to calculate the average velocity from the definition of average velocity, formula (U_A_M 1), . To use this formula, the total displacement and the total time need to be calculated. The procedure presented below follows this approach. The total displacement can be easily obtained by just adding the two partial displacements,


Accordingly with the notation used on the drawing on the left, this formula can be utilized for solving for the time t1, time taken by the UFO to travel the first 700 m, and the time t2, time taken by the UFO to travel the second 600 m.

Solving for the time in formula (U_A_M 1),


The following results are obtained


From the previous results, the total time of the trip is calculated,


Now, the results for the total displacement and for the total time of the trip can be putted together into the definition of average velocity with the purpose of obtaining the UFO average velocity for the entire trip,

Notice that this result is different than the result obtained using formula (U_A_M 2),

Only the result obtained using the definition of average velocity is the correct one.


by Luis F. Sez, Ph. D.    Comments and Suggestions: LSaez@dallaswinwin.com