Non-constant acceleration using du/dt = alpha - Beta(u sub t)

[Skoon86]

I'm trying to understand the process for solving the maximum acceleration of a vehicle given two known speeds and their times, assuming the acceleration in non-constant and in the form of du/dt = α - βu_t.

The problem is stated as a driver of a vehicle on a level road determined she could increase her speed from a stop to 73 fps in 35 seconds and from stop to 100 fps in 95 seconds. Assume the acceleration in the form (noted above), determine the maximum acceleration of the vehicle.

The next question is if the vehicle above 3 is traveling at a speed of 40 mph, how long will it take after the driver starts accelerating for the vehicle to achieve a speed of 45 mph?

 

[JeffM]

Since we have no idea what your variables mean, your question is meaningless. Is u acceleration? What is u_t?

“The vehicle above three” means what? A number is an idea and cannot be below a vehicle.

Are you studying differential equations? What have you tried? What is the exact and complete wording of the actual problem?

 

[Skoon86]

U_t is vehicle speed in ft/sec.

“The vehicle above three” should have read "... if the vehicle in the above problem is traveling at a speed of 40 mph..."

These problems are from a Traffic and Highway Engineering course and we're looking at Traffic Operations. The exact and complete wording of the problem(s) are:

a) The driver of a vehicle on a level road determined that she could increase her speed from rest to 50 mph in 34.8 sec and from rest to 65 mph in 94.8 sec. If it can be assumed that the acceleration of the vehicle takes the form

, determine the maximum acceleration of the vehicle.
b) If the vehicle in Problem a) is traveling at a speed of 40 mph, how long will it take after the driver starts accelerating for the vehicle to achieve a speed of 45 mph?

My apologies, it's been over 30 years since I've dealt with this complex math.