Many car companies are performing research on collision avoidance systems. a small prototype applies engine braking that decelerates the vehicle according to the relationship a = − k √ t , where a and t are expressed in m/s² and seconds, respectively.
the vehicle is traveling at 20 m/s when its radar sensors detect a stationary obstacle. knowing that it takes the prototype vehicle 4 seconds to stop, determine; (a) expressions for its velocity and position as a function of time, (b) how far the vehicle traveled before it stopped.

Explanation:

In this case acceleration is defined as:

,

where k is a constant to be found.

To find the expressions for velocity and position as a function of time you must integrate the expression above for acceleration two times.

Initial conditions and boundary conditions are defined with the rest of the data as:

First integration is equal to:

The boundary condition and initial condition can be used to calculate and :

With this expression for velocity is defined as:

The same can be done to get to expression for position:

To get the total distance traveled you can integrate the velocity expression from time=0 sec to time=4 sec:

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