Question

The velocity of a body is given by the equation $v = 6 - 0 \cdot 02t$, where $t$ is the time taken. What does the body undergo?
A) Uniform retardation of $0 \cdot 02{\text{m}}{{\text{s}}^{ - 2}}$
B) Uniform acceleration of $0 \cdot 02{\text{m}}{{\text{s}}^{ - 2}}$
C) Uniform retardation of $0 \cdot 04{\text{m}}{{\text{s}}^{ - 2}}$
D) Uniform acceleration of $0 \cdot 04{\text{m}}{{\text{s}}^{ - 2}}$

Answer 1

Newton devised three equations of motion which help us to calculate the displacement, the velocity or acceleration of a body in translational motion. Newton’s second equation of motion describes the velocity of a body and can be used to obtain the acceleration of the given body. A negative acceleration implies retardation.

Formula used:
-Newton’s second equation of motion is given by, $v = u + at$ where $v$ is the final velocity of the body, $u$ is its initial velocity, $a$ is the body’s acceleration and $t$ is the time taken.

Complete step by step answer.
Step 1: State the given equation of the velocity of the body.
The given equation of the velocity of the body is $v = 6 - 0 \cdot 02t$ ---------- (1) where $t$ is the time taken.
Step 2: Express Newton’s second equation of motion.
Newton’s second equation of motion is given by, $v = u + at$ ---------- (2) where $v$ is the final velocity of the body, $u$ is its initial velocity, $a$ is the body’s acceleration and $t$ is the time taken.
Equations (1) and (2) describe the velocity of a body. So on comparing these two equations, we find $u = 6{\text{m}}{{\text{s}}^{ - 1}}$ and $a = - 0 \cdot 02{\text{m}}{{\text{s}}^{ - 2}}$ .
As the obtained acceleration of the body is negative, the body will be undergoing uniform retardation of magnitude $a = 0 \cdot 02{\text{m}}{{\text{s}}^{ - 2}}$.

So the correct option is A.

Note: Alternate method
The given equation of the velocity of the body is $v = 6 - 0 \cdot 02t$ -------- (A).
The acceleration of the body is defined as the rate at which the velocity of a body changes i.e., $a = \dfrac{{dv}}{{dt}}$
So taking the derivative of equation (A) will give us the acceleration of the body.
Thus we have $a = \dfrac{{dv}}{{dt}} = \dfrac{d}{{dt}}\left( {6 - 0 \cdot 02t} \right) = - 0 \cdot 02$
Thus we obtain the acceleration of the body as $a = - 0 \cdot 02{\text{m}}{{\text{s}}^{ - 2}}$. However, the negative sign indicates that it is actually retardation or deceleration. Hence the correct option is A.