The distance s of a particle moving along a straight line from a fixed-pointO on the line at time t seconds after start is given by x = (t – 1)2(t – 2)2. What will be the distance of the particle from O when its velocity is zero?

The distance s of a particle moving along a straight line from a fixed-pointO on the line at time t seconds after start is given by x = (t – 1)2(t – 2)2. What will be the distance of the particle from O when its velocity is zero? Correct Answer 4/27 units

Let v be the velocity of the particle at time t seconds after start (that is at a distance s from O). Then, v = ds/dt = d/dt Or v = (t – 2)(3t – 4) Clearly, v = 0, when (t – 2)(3t – 4) = 0 That is, when t = 2 Or 3t – 4 = 0 i.e., t = 4/3 Now, s = (t – 1)(t – 2)2 Therefore, when t = 4/3, then s = (4/3 – 1)(4/3 – 2)2 = 4/27 And when t = 2, then s =(2 – 1)(2 – 2)2 = 0 Therefore, the velocity of the particle is zero, when its distance from O is 4/27 units and when it is at O.