A racing car of mass m travels along a straight road. While travelling, the car is subject to a constant frictional resistance force of magnitude a*m (a, a positive constant) and an air resistance force of magnitude k*(v^2), where v is its speed. The car engine can provide a constant propelling force of magnitude b*m (b a positive constant), which can bring the racing car to a terminal speed u'

a) show that k=m*(b-a)/(u')^2

b) hence show that du/dt = (b-a) (1-(u/u')^2)

c) the car starts from rest at time t=0. Show that the car reaches speed 0.5u' at time t=t1/2,

where t1/2=(u'*ln3)/2(b-a)

d) at time t=t1/2, the engine is switched off. Show that the car comes to rest at time

t1/2 + (u'/sqrt(a*(b-a)))*(tan^-1(sqrt(b-a)/4a))

I have done the first three parts. I have done the first part by considering what is the force balance on the car while is travelling with constant speed v'. I have done the second part by considering thatF=m*a=>a=f/m I have done also the third part by integrating from 0 to v'/2 of (1/(dv/dt)) dv

I have stucked at part d. I realize that when the engine is switched off the propelling force of the engine stops to act on the car and then we only have the air resistance force and the frictional resistance, two forces that will decrease the speed of the car until the car comes to rest.

Any help would be appreciated. Thanks in advance.

Kind regards!