A particle moves in a straight line under the action of a retarding force is the initial speed is U and the retarding force is MKV^3 where M is mass V is the velocity at any instant K is constant show that 1/V-1/U=KX,X being the distance covered before its speed reduced to V

Keepin in mind that the velocity v is in opposite direction of the force F, and direction of force to be negative

we solve the problem

thus

we get

now using the initial condition at x=0, v=u we get c = 1/u and hece we get the desired result


Part 2:
using the previous result and v = dx/dt we have the following integral equation

and we get

again by initial condition that at t=0 x=0 we find the value of the constant to be 0 and hence we get the desired result