A particle moves on a straight line with constant acceleration of magnitude a and reverses its direction once. Consider all possible time intervals of duration τ which contain the turning instant. The minimum possible actual distance travelled by the particle during such an interval is:
Options

Question

A particle moves on a straight line with constant acceleration of magnitude a and reverses its direction once. Consider all possible time intervals of duration τ which contain the turning instant. The minimum possible actual distance travelled by the particle during such an interval is:

Options

G1 Physics · Accepted Answer

Let the turning instant divide the interval into durations t and τ−t. Since velocity is zero at the turning point, distance travelled before and after it is s = (a/2)t² + (a/2)(τ−t)². So s = (a/2)[t²+(τ−t)²]. Minimize with respect to t: ds/dt = a(2t−τ)=0 ⇒ t=τ/2. Hence sₘᵢₙ = (a/2)[τ²/4+τ²/4] = aτ²/4. Therefore option B is correct.