The motion of a particle along a straight line is described by the equation x = 8 + 12t - t³, where x is in me

Question

The motion of a particle along a straight line is described by the equation x = 8 + 12t - t³, where x is in metre and t is in second. The retardation of the particle when its velocity becomes zero is:

Accepted Answer

1.  **Find Velocity ($v$):** Velocity is defined as the rate of change of position with respect to time ($v = \frac{dx}{dt}$) .
    Given $x = 8 + 12t - t^3$.
    $$ v = \frac{d}{dt}(8 + 12t - t^3) = 0 + 12 - 3t^2 $$
2.  **Find Time ($t$) when Velocity is Zero:**
    Set $v = 0$:
    $$ 12 - 3t^2 = 0 $$
    $$ 3t^2 = 12 \implies t^2 = 4 \implies t = 2 	ext{ s} $$
3.  **Find Acceleration ($a$):** Acceleration is defined as the rate of change of velocity with respect to time ($a = \frac{dv}{dt}$) .
    $$ a = \frac{d}{dt}(12 - 3t^2) = -6t $$
4.  **Calculate Retardation:**
    At $t = 2$ s, acceleration $a = -6(2) = -12 	ext{ ms}^{-2}$.
    Retardation is the negative of acceleration (or the magnitude of deceleration). Therefore, retardation is $12 	ext{ ms}^{-2}$.

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