The motion of a particle along a straight line is described by equation :38 12x t t   where x is in metre and t in second. The retardation of the particlewhen its velocity becomes zero is

The given equation is x = 38t - 12t². This is an equation of motion where x is the displacement and t is the time.

Step 1: Find the velocity The velocity v of the particle at any time t is given by the derivative of the displacement x with respect to time t. So, differentiate x = 38t - 12t² with respect to t to get the velocity.

v = dx/dt = 38 - 24t

Step 2: Find the time when velocity is zero Set v = 0 and solve for t.

0 = 38 - 24t 24t = 38 t = 38/24 = 1.583 seconds

Step 3: Find the acceleration The acceleration a of the particle at any time t is given by the derivative of the velocity v with respect to time t. So, differentiate v = 38 - 24t with respect to t to get the acceleration.

a = dv/dt = -24 m/s²

Since the acceleration is negative, this is actually a retardation (or deceleration). So, the retardation of the particle when its velocity becomes zero is 24 m/s².