Question

If the displacement \(S\) of a particle travelling along a straight line in \(t\) seconds is given by \[ S = 2t^3 + 2t^2 - 2t - 3, \] then the time taken (in seconds) by the particle to change its direction is?

• A. \(\frac{1}{3}\)
• B. 2
• C. 3
• D. \(\frac{1}{2}\)

Hint:

Change in direction occurs when velocity changes sign, find roots of velocity function.

Answer 1

The Correct Option is A

Velocity \(v = \frac{dS}{dt} = 6t^2 + 4t - 2\). Set velocity zero for change in direction: \[ 6t^2 + 4t - 2 = 0. \] Solve quadratic: \[ t = \frac{-4 \pm \sqrt{16 + 48}}{12} = \frac{-4 \pm 8}{12}. \] Positive root: \[ t = \frac{1}{3}. \]