Question

The velocity of a body at the end of $4\;{\rm{seconds}}$ is
$26\;{\rm{m}}{{\rm{s}}^{ - 1}}$, at the end of $12\;{\rm{seconds}}$ is $58\;{\rm{m}}{{\rm{s}}^{ - 1}}$ and at the end of $22\;{\rm{seconds}}$ is
$98\;{\rm{m}}{{\rm{s}}^{ - 1}}$. The body is moving with:-
(A) Uniform acceleration
(B) Uniform speed
(C) Uniform velocity
(D) Uniform displacement

Answer 1

Hint: In this question, the body's velocity is given to us at a different time interval, so we will use the equation of motion at a different time interval. The various equations will give us information about the acceleration, speed, velocity, and displacement of the body.

Complete step by step answer:
It is given to us that velocity of the body at the end of $4\;{\rm{second}}$ is $26\;{\rm{m}}{{\rm{s}}^{ - 1}}$, at the end of $12\;{\rm{seconds}}$ is $58\;{\rm{m}}{{\rm{s}}^{ - 1}}$ and at the end of $22\;{\rm{second}}$ is $98\;{\rm{m}}{{\rm{s}}^{ - 1}}$. So, write the equation of motion for the condition of the body after $4\;{\rm{seconds}}$.
$\Rightarrow v = u + at$
Here $v$ is the velocity of the body after $4\;{\rm{seconds}}$ and $u$ is the initial velocity of the body, $a$ is the body's acceleration in the first condition, and $t$ is the time.
Substitute the values in the above equation.
Therefore, we get
$\Rightarrow 26\;{\rm{m}}{{\rm{s}}^{ - 1}} = 0 + a\left( {4\;{\rm{s}}} \right)$ …… (1)
Write the equation of motion for the second condition when body’s velocity is $\Rightarrow 58\;{\rm{m}}{{\rm{s}}^{ - 1}}$ at the end of $12\;{\rm{seconds}}$. So,
$\Rightarrow v = u + at$
Here we will substitute the values in the above equation that are given in the second condition. So,
$\Rightarrow 58\;{\rm{m}}{{\rm{s}}^{ - 1}} = u + a\left( {12\;{\rm{s}}} \right)$ …… (2)
Write the equation of motion for the third condition when body’s velocity is $\Rightarrow 98\;{\rm{m}}{{\rm{s}}^{ - 1}}$ at the end of $22\;{\rm{seconds}}$. So,
$\Rightarrow v = u + at$
Here we will substitute the values in the above equation that are given in the third condition. So,
$\Rightarrow 58\;{\rm{m}}{{\rm{s}}^{ - 1}} = u + a\left( {12\;{\rm{s}}} \right)$ …… (3)
Use equations (1) and (2) to determine the acceleration of the body.
Therefore, we get
$
\Rightarrow 26\;{\rm{m}}{{\rm{s}}^{ - 1}} = \left( {58\;{\rm{m}}{{\rm{s}}^{ - 1}} - a\left( {12\;{\rm{s}}} \right)} \right) + a\left( {4\;{\rm{s}}} \right)\\
\Rightarrow a\left( {12\;{\rm{s}}} \right) - a\left( {4\;{\rm{s}}} \right) = 58\;{\rm{m}}{{\rm{s}}^{ - 1}} - 26\;{\rm{m}}{{\rm{s}}^{ - 1}}\\
\Rightarrow a\left( {8\;{\rm{s}}} \right) = 32\;\;{\rm{m}}{{\rm{s}}^{ - 1}}\\
\Rightarrow a = 4\;{\rm{m}}{{\rm{s}}^{ - 2}}
$
We use equations (2) and (3) to determine the body's acceleration at different times.
$
\Rightarrow 58\;{\rm{m}}{{\rm{s}}^{ - 1}} = \left( {98\;{\rm{m}}{{\rm{s}}^{ - 1}} - a\left( {22\;{\rm{s}}} \right)} \right) + a\left( {{\rm{12}}\;{\rm{s}}} \right)\\
\Rightarrow a\left( {22\;{\rm{s}}} \right) - a\left( {12\;{\rm{s}}} \right) = 98\;{\rm{m}}{{\rm{s}}^{ - 1}} - 56\;{\rm{m}}{{\rm{s}}^{ - 1}}\\
\Rightarrow a\left( {10\;{\rm{s}}} \right) = 40\;\;{\rm{m}}{{\rm{s}}^{ - 1}}\\
\Rightarrow a = 4\;{\rm{m}}{{\rm{s}}^{ - 2}}
$
Therefore, the acceleration values are the same at two different times, so the body's acceleration is uniform, and option (A) is correct.

Note: we used the first equation of motion in the solution because information about the displacement is not given in the question. The question only gives information about velocity and time. The second and third equation of motion consists of displacement terms, so it's become difficult for us to determine the correct answer.