Question

A car starts from rest, moves with an acceleration $6m{{s}^{-2}}$ then decelerates with $2m{{s}^{-2}}$ for some time to come to rest. If the total time taken is 16 s, then the instantaneous time of maximum velocity is
A. 8s
B. 12s
C. 4s
D. 10s

Answer 1

The motion of a body is explained very nicely using the equations of motion. These equations are used to estimate the theoretical values of the initial and final velocity, acceleration, time, distance which the body must achieve while performing the work done. Three equations show the relation between these values and are termed as the equation of motion.
As per the given data
Initial acceleration is $6m{{s}^{-2}}$
Total time is taken to before the deceleration is $16s$
Deceleration of the body is with a value of $2m{{s}^{-2}}$

Formula used:
$v=u+at$

Complete step-by-step answer:
According to the condition of the acceleration and the deceleration of the car. The whole journey of the car can be divided into two cases,
Case 1: (When the body is accelerating from rest)
In a moving body from a rest position, the initial velocity of the body is zero. The final velocity can be found by using the 1st equation of motion.
Using the 1st equation of motion we can say that,
$v=u+at$
By putting the values accordingly,
$\begin{aligned} & v=0+6t \\\ & \Rightarrow v=6t \\\ \end{aligned}$
Thus the maximum velocity that can be achieved by the body is $6t$ .

Case 2: (When the body starts to decelerate after 16s)
When a body undergoes retarding motion the final value of the velocity is zero. Here as mentioned in the question the maximum acceleration is achieved in a total time of $16s$. So the instantaneous time of the maximum acceleration can be found by using the 1st equation of motion.
Using the 1st equation of motion we can say that to achieve zero velocity, the acceleration is $2m{{s}^{-2}}$ in the opposite direction.
So, $v=u+at$
$\begin{aligned} & 0=6t-2(16-t) \\\ & \Rightarrow 8t=32 \\\ & \Rightarrow t=4s \\\ \end{aligned}$
So, The instantaneous time for the maximum velocity is calculated to be equal to $4s$.
Thus the correct option which shows the true value of the asked instantaneous time for the maximum velocity is Option C.

So, the correct answer is “Option C”.

Note: The maximum velocity attained by the body is the final velocity that the body can achieve during acceleration. This value will be the initial value of velocity when the car starts to decelerate. The instantaneous time of deceleration will be less than the total time is taken by the body to achieve maximum acceleration.