Question

Which are of the following represents uniformly accelerated motion:
A. \[x = \sqrt {\dfrac{{t - a}}{b}} \]
B. \[x = \dfrac{{t - a}}{b}\]
C. \[t = \sqrt {\dfrac{{x - a}}{b}} \]
D. \[x = \sqrt t + a\]

Answer 1

Hint: Recall the conditions which are necessary for a motion to be uniformly accelerated. Check each option one by one, find the acceleration of each and observe how it is depended on time and whether it represents uniformly accelerated motion or not. Use the formula for acceleration.

Complete step by step answer:
Given to find which of the following options represents uniformly accelerated motion.
At first let us know what a uniformly accelerated motion is. Uniformly accelerated motion means the acceleration of the body remains constant, that is the acceleration does not change with time or it is independent of time. So, we can write
\[\dfrac{{da}}{{dt}} = 0 \\
\Rightarrow a = \,{\text{constant}} \]
Acceleration can also be written in terms of position, as acceleration is the rate change in velocity and velocity is rate of change in position. So, we can write
\[a = \dfrac{{dv}}{{dt}} = \dfrac{d}{{dt}}\left( {\dfrac{{dx}}{{dt}}} \right) \\
\Rightarrow a = \dfrac{{{d^2}x}}{{d{t^2}}} \]
For uniformly accelerated motion, \[a\] is constant so, \[\dfrac{{{d^2}x}}{{d{t^2}}}\] should be a constant.

Now, for each option we find out the value of \[\dfrac{{{d^2}x}}{{d{t^2}}}\]
Option A: \[x = \sqrt {\dfrac{{t - a}}{b}} \]
\[ \Rightarrow {x^2} = \dfrac{{t - a}}{b}\]
We differentiate \[x\] with respect to \[t\] and we get
\[2\dfrac{{dx}}{{dt}} = \dfrac{1}{b} - \dfrac{a}{b} \\
\Rightarrow \dfrac{{dx}}{{dt}} = \dfrac{1}{2}\left( {\dfrac{1}{b} - \dfrac{a}{b}} \right) \]
Here , the rate of change in position is constant which means there is velocity is zero and hence acceleration is also zero.

Option B: \[x = \dfrac{{t - a}}{b}\]
Differentiating \[x\] with respect to \[t\] we get
\[\dfrac{{dx}}{{dt}} = \left( {\dfrac{1}{b} - \dfrac{a}{b}} \right)\]
Here too the rate of change in position is constant which means there is velocity is zero and hence acceleration is also zero.

Option C: \[t = \sqrt {\dfrac{{x - a}}{b}} \]
\[\Rightarrow {t^2} = \dfrac{{x - a}}{b} \\
\Rightarrow x = a + b{t^2} \]
Differentiating \[x\] with respect to \[t\] we get
\[\dfrac{{dx}}{{dt}} = 2bt\]
Again differentiating we get,
\[\dfrac{{{d^2}x}}{{d{t^2}}} = 2b\]
Here, \[\dfrac{{{d^2}x}}{{d{t^2}}}\] is constant and we have \[a = \dfrac{{{d^2}x}}{{d{t^2}}}\] that means the acceleration is constant, so it is an uniformly accelerated motion.

Option D: \[x = \sqrt t + a\]
Differentiating \[x\] with respect to \[t\] we get
\[\dfrac{{dx}}{{dt}} = \dfrac{1}{2}\dfrac{1}{{\sqrt t }}\]
Again differentiating we get,
\[\dfrac{{{d^2}x}}{{d{t^2}}} = \dfrac{1}{4}\dfrac{1}{{{t^{3/2}}}}\]
Here, we can see the \[\dfrac{{{d^2}x}}{{d{t^2}}}\] is dependent on time, that is acceleration is dependent on time. So, it is not an uniformly accelerated motion.

Hence, the correct answer is option C.

Note: A uniform motion is a motion which does not change with time. If a body is said to be moving with uniform velocity that means the velocity of the body is not changing with time or there is no acceleration. If a body is said to be moving with non-uniform velocity then it means there is some acceleration and the velocity of the body is changing with time.