A particle moving in a straight line traverses a distance x in time t. If t = x2/2 + x, then which one is correct?

A particle moving in a straight line traverses a distance x in time t. If t = x2/2 + x, then which one is correct? Correct Answer The retardation of the particle is the cube of its velocity

We have, t = x2/2 + x Therefore, dt/dx = 2x/2 + 1 = x + 1 Thus, if v be the velocity of the particle at time t, then v = dx/dt = 1/(dt/dx) = 1/(x + 1) = (x + 1)-1 Thus dv/dt = d((x + 1)-1)/dt = (-1)(x + 1)-2 d(x + 1)/dt = -1/(x + 1)2 * dx/dt As, 1/(x + 1) = dx/dt, So, -(dx/dt)2(dx/dt) Or dv/dt = -v2*v   = -v3 We know, dv/dt = acceleration of a particle. As, dv/dt is negative, so there is a retardation of the particle. Thus, the retardation of the particle = -dv/dt = v3 = cube of the particle.