Question

A particle moves in a straight line with a constant acceleration. It changes its velocity from $10m{s^{ - 1}}$ to $20m{s^{ - 1}}$ while passing through a distance $135m$in $t$ second. The value of $t$ is.
(A) 12
(B) 9
(C) 10
(D) 1.8

Answer 1

We know that equations of motion are a set of equations that are derived to assess the motion of a body and the characteristics of its motion, given that the body is experiencing constant acceleration. These motions apply only to bodies moving linearly or experiencing a linear motion with respect to a reference point. The equations of motions are:
$v = u + at$
$s = ut + \dfrac{1}{2}a{t^2}$
${v^2} - {u^2} = 2as$
These equations are used to find the magnitude of the vector quantities that specify motion of a body. Only the magnitudes are needed because motion is along a straight line and hence position, velocity and acceleration are collinear.
Formula used: We will be using the formulas from the equations of motion, $v = u + at$ where $v$ is the final velocity of the body, $u$ is the initial velocity, $a$ is the uniform acceleration experienced by the body, and $t$ is the time taken.
We will also be using the formula $s = ut + \dfrac{1}{2}a{t^2}$ where $s$ is the displacement covered by the body.

Complete Step by Step answer:
We know that when a body is executing uniform acceleration the motion of the body and its magnitude can be specified by the equations of motion (also known as the SUVAT equations). From the problem we know that the particle is executing uniform acceleration in a straight line. The velocity changes from $u = 10m{s^{ - 1}}$ to $v = 20m{s^{ - 1}}$ through a distance $s = 135m$ .
We know from the equations of motion that,
$v = u + at$
$\Rightarrow a = \dfrac{{v - u}}{t}$
Substituting the values, we know,
$a = \dfrac{{20 - 10}}{t}$
$\Rightarrow a = \dfrac{{10}}{t}$
Using these values in the equation of motion, $s = ut + \dfrac{1}{2}a{t^2}$ .
$135 = \left( {10} \right)t + \dfrac{1}{2}\left( {\dfrac{{10}}{t}} \right){t^2}$
$135 = 10t + 5t$
Solving for $t$ we get,
$135 = 15t$
$\Rightarrow t = \dfrac{{135}}{{15}} = 9s$

Thus, the time taken for the motion of the particle is $t = 9s$ . Hence the correct answer is option B.

Note: We can also use the equation, ${v^2} - {u^2} = 2as$ first to find the value of $a = 1.1m{s^{ - 2}}$. And then we need to use the equation $v = u + at$ to find the time taken $t = 9.09s \approx 9s$.