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A body $A$ moves with a uniform acceleration $a$ and zero initial velocity. Another body $B$ , starts from the same point moves in the same direction with a constant velocity $v$ . The two bodies meet after a time $t$ . The value of $t$ is
(A) $2v/a$
(B) $v/a$
(C) $v/2a$
(D) $\sqrt {v/2a} $

Last updated date: 20th Jun 2024
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Hint: We will use the equations of motion. Out of all the equations, we will select the appropriate equation connecting all the parameters. Finally, we will find the appropriate relation.

Formulae Used: \[s = ut + {\text{ }}\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/
 \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} {\text{ }}a{t^2}\]
Where, $s$ is the displacement of the body, $u$ is the initial velocity of the body, $t$ is the time taken and $a$ is the acceleration of the body.

Step By Step Solution
For the body $A$ ,
\[u = 0\]
\[a = a\]
Thus, the formula turns out to be,
\[s = {\text{ }}\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/
 \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} {\text{ }}a{t^2} \cdot \cdot \cdot \cdot (1)\]
For the body $B$ ,
\[u = v\]
\[a = 0\]
Thus, the formula turns out to be,
\[s = vt \cdot \cdot \cdot \cdot (2)\]
Equating $(1)$ and$(2)$, we get
\[\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/
 \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} {\text{ }}a{t^2} = vt\]
\[ \Rightarrow t = 2v/a\]

Hence, the answer is (a).

Additional Information: The equations of motions helps us to connect all the parameters related to motion such as displacement, distance, speed, velocity time and acceleration. The usage of this equation depends on the given situation and the required parameter to evaluate.
The equations are:
$v = u + at$
Here, the final parameter to find the final velocity (/speed) when the initial velocity (speed), acceleration and time are given.
\[s = ut + {\text{ }}\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/
 \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} {\text{ }}a{t^2}\]
Here, the final parameter to find is the displacement (in some cases distance), when the initial velocity (/speed), acceleration and time are known.
\[{v^2} - {u^2} = 2as\]
Here, the motive is to relate all the parameters.
Thus, every equation has its own purpose at the same time restrictions to be used. These equations come in very handy for solving any type of situation in motion.

Note: We took the displacement same for both the bodies. This is because they started from the same position.