Question

A particle moves along x-axis obeying the equation x = t (t – 1)(t – 2), where x (in metres) is the position of the particle at any time t (in seconds). The displacement when the velocity of the particle is zero, is 

Options

• `2/(3sqrt(3)) m, 2/(3sqrt(3)) m`

• `5/(3sqrt(3)) m, 5/(3sqrt(3)) m`

• – 3 m, 3 m

• – 5 m, 5 m

Answer 1

`2/(3sqrt(3)) m, 2/(3sqrt(3)) m`

Explanation:

Displacement x = t(t – 1)(t – 2)  ......(i)

∴ x = t³ – 2t² – t² + 2t

Velocity v = `(dx)/(dt)`

∴ v = 3t – 6t + 2

For v = 0, we have

3t² – 6t + 2 = 0  ......(ii)

Equation (ii) is quadratic in t

Solving we get,

t = `(1 + 1/sqrt(3))` or t = `(1 - 1/sqrt(3))` ......(iii)

Now; x = t(t – 1) (t – 2)  ......[From (i)]

Substitute equation (iii) in equation (i)

x = `(1 + 1/sqrt(3)) [1 + 1/sqrt(3) - 1] [1 + 1/sqrt(3) - 2]`

Solving we get,

x = `2/(3sqrt(3) m`

Similarly, By substituting the other value of t from equation (iii) in equation (i) and solving we get,

x = `2/(3sqrt(3)) m`