One model for the behavior of a car when it accelerates to passanother vehicle i

Question

One model for the behavior of a car when it accelerates to passanother vehicle is to assume that it does this while the enginedelivers power at a constant rate. Under these circumstances themagnitude of the car's acceleration is inversely proportional toits speed. Let's say that you are driving such a car thataccelerates according to

ax = B / vx

where B = 103.0 m2/s3 is a parameterthat is related to the ratio of your car's power to its weight, andvx is your car's velocity in thex-direction, in m/s. Assume that you are initiallytravelling with a speed of 23.5 m/s. At t = 0 you step onthe gas pedal. The car performs a constant-power acceleration untilyou reach a speed of 42.3 m/s. What is the time intervalDt needed to make this changeof speed?

Explanation / Answer

you know:      dv = adt      and       a = B/v     so put these together: .          dv = ( B / v ) dt     or          v dv   = B dt . integrate both sides: .         (1/2) ( vfinal2 - v initial2 ) = B t .    (1/2) ( 42.32 - 23.52 ) = 103.0 t .     t =   6.005 seconds

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