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

CONCEPT: 

ACCELERATION & DECELEARATION:

• Acceleration is the measure of the rate of change of velocity with respect to time, the value of acceleration can be both positive and negative.  

​ \(a = \frac{{v - u}}{t}\)                                                                                                                                                                                                                  Where v= Final velocity, u =Initial velocity and t = time 

•  Deceleration is the negative rate of change of velocity.        

CALCULATION:

Given - x = 8 + 12t - t3

• As we know, velocity is defined as the rate of change of displacement.

 \(⇒ V = \frac{{dx}}{{dt}} = \frac{{d(8 + 12t - {t^3})}}{{dt}} = 12 - 3{t^2}\)  

When V = 0, then

⇒ 12 - 3t² = 0

⇒ 12 = 3t²

⇒ t = 2 sec

• The retardation is given by

 \(⇒ a = - \frac{{dV}}{{dt}} = - \frac{{d(12 - 3{t^2})}}{{dt}} = - ( - 6t) = 6\times 2 = 12m{s^{ - 2}}\)