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A body moves for a total of 9 seconds starting from rest with uniform acceleration and uniform retardation which is twice the value of acceleration and then stops. The duration of uniform acceleration is:
(A) $3s$
(B) $4.5s$
(C) $5s$
(D) $6s$

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Last updated date: 19th Apr 2024
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Answer
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Hint: We need to consider two cases here. The first case being, a body that moves with a certain acceleration, for some time and the second case where the same body has retardation.
Formula Used: The formulae used in the solution are given here.
$v = u + at$ where $v$ is the final velocity, $u$ is the initial velocity, $a$ is the acceleration and $t$ is the time taken.

Complete Step by Step Solution: Let the acceleration of the body be $a$ and the retardation be $b$.
By Newton’s law of motion, $v = u + at$ where $v$ is the final velocity, $u$ is the initial velocity, $a$ is the acceleration and $t$ is the time taken.
It has been given in the question, that, a body moves for a total of 9 seconds starting from rest with uniform acceleration and uniform retardation which is twice the value of acceleration and then stops.
Thus, according to the question, $2a = b$.
For a body that moves with acceleration $a$, for time $t$ seconds,
$v = 0 + at$ where initial velocity $u = 0$ and $v$ is the final velocity.
When the same body retards with acceleration $2a$, for time $t'$ seconds and then ends at rest. Thus final velocity of the body is zero.
Here, initial velocity, $u = at$, the same as the final velocity of the previous case.
Since, $v = 0$, and acceleration is $ - 2a$.
So, $0 = at - 2at'$.
From here, $t = 2t'$.
Given that, $t + t' = 9s$. Thus,
$t' + 2t' = 9$
$ \Rightarrow t' = 3s$
and $t = 6s$.
The duration of uniform acceleration is $t' = 3s$.

Hence, the correct answer is Option A.

Note: Retardation is nothing but a negative acceleration. The velocity of the body may either increase or decrease. The change in velocity is known as acceleration. If the velocity of the body increases, acceleration is said to be positive. Similarly, if the velocity decreases, the acceleration is said to be negative.
The train reaching the station slows down, in this case, we can say that the train is retarding. Retardation is acceleration with a negative sign. Or the negative value of acceleration shows that the velocity of a body is decreasing.