Acceleration?

Question

The driver of a vehicle on a level road determined that she could increase her speed from rest to 50 mi/h in 34.8 seconds and from rest to 65 mi/h in 94.8 seconds. If it can be assumed that the acceleration of the vehicle takes the form du/dt = alpha - (beta)u determine the maximum acceleration in each case. Where alpha and beta are constants and u is the vehicle speed in ft/sec.

Answer 1

If beta > 0, then alpha is the maximum acceleration. Can you find alpha and beta?
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Ideas:
du/(alpha-beta*u) = dt
(-1/beta)ln|(alfa-beta*u)/alfa| = t
=> (-1/beta)ln|(alfa-beta*50)/alfa| = 34.8......(1)
and (-1/beta)ln|(alfa-beta*65)/alfa| = 94.8......(2)
Solve (1) and (2) for alpha and beta.

Answer 2

If we call [A-Bu] = V, we can write
du/V = dt. Since dV=-Bdu, we can multiply both sides by -B to get -Bdu/V=-Bdt or dV/V=-Bdt. This integrates to ln(A-Bu)=-Bt+c1. Since u=0 when t=0, c1 can be evaluated. Then we can use the resulting equation twice with the two end points of velocity and time to determine the unknown constants A and B.