Question

The equation of motion of a particle starting from rest along a straight line is \( x = t^3 - 3t^2 + 5 \). The ratio of the accelerations after 5 s and 3 s will be:

• A. 2
• B. 3
• C. 4
• D. 5

Hint:

To find acceleration from a position-time equation, differentiate twice. Use values of \( t \) directly after finding the general expression.

Answer 1

The Correct Option is A

Step 1: Differentiate position to find acceleration.
Given: \[ x = t^3 - 3t^2 + 5 \] Velocity: \[ v = \frac{dx}{dt} = 3t^2 - 6t \] Acceleration: \[ a = \frac{dv}{dt} = \frac{d^2x}{dt^2} = 6t - 6 \]
Step 2: Find accelerations at \( t = 5 \) and \( t = 3 \).
At \( t = 5 \): \[ a_1 = 6(5) - 6 = 30 - 6 = 24 \] At \( t = 3 \): \[ a_2 = 6(3) - 6 = 18 - 6 = 12 \]
Step 3: Take the ratio.
\[ \frac{a_1}{a_2} = \frac{24}{12} = 2 \]