At the scene of an accident, there are skid marks which are 200 feet long showing where a car put on the brakes. The driver swears that he was obeying the speed limit. If we assume that the car braked with a constant deceleration of 16ft/sec^2, we can determine whether or not the driver is telling the truth. how fast was he going when he applied the brakes?

He was going at 54.5mph

Explanation:

Measuring distances in feet and time (starting with the brakes being applied) in seconds,

we have a(t) = −16, s(0) = 0, and v(T) = 0, s(T) = 200,

where T is the time of braking.

So Finding antiderivatives, v(t) = −16t + A and so s(t) = −8t²+ At +B. Since s(0) = 0,

we have B = 0. Since v(T) = 0,

we have T = A /16.

Putting this into s() gives

200 = −8(A/ 16)² +A· (A /16) = A²/32.

This gives A2 = 6,400, so A = 80 = v(0). That is, the initial speed was 80 ft/s ≈ 54.5 mph.