Question

A particle starting with certain initial velocity and uniform acceleration covers a distance of 12 m in the first 3 s and a distance of 30 m in the next 3 s. The initial velocity of the particle is
(A) $ 3m{s^{ - 1}} $
(B) $ 2.5m{s^{ - 1}} $
(C) $ 2m{s^{ - 1}} $
(D) $ 1.5m{s^{ - 1}} $
(E) $ 1m{s^{ - 1}} $

Answer 1

Hint: This could be simply solved by breaking the total distance into two parts and applying the basic formula of time. Here, we would be using the equation of motion.

Formula used Here, we will use the equation of motion:
 $ s = ut + \dfrac{1}{2}a{t^2} $
Here, $ s $ is the distance travelled
 $ u $ is the initial velocity
 $ t $ is the travel time
 $ a $ is the acceleration of the particle.

Complete step by step answer
We will start by considering the acceleration to be $ a $ ,
Given,
 $ {{\text{s}}_{\text{1}}}{\text{ = 12m,}}{{\text{t}}_{\text{1}}}{\text{ = 3s}} $
Similarly, the distance to be travelled in second time period:
Given
 $ {{\text{s}}_2}{\text{ = 30 + 12m, }}{{\text{t}}_2}{\text{ = 3 + 3s}} $ $ \begin{gathered}
  {{\text{v}}_{\text{1}}}{\text{ = 10kmph}} \\
  {{\text{t}}_{\text{1}}}{\text{ = }}\dfrac{{{\raise0.7ex\hbox{ $ {\text{d}} $ } \!\mathord{\left/
 {\vphantom {{\text{d}} {\text{3}}}}\right.}
\!\lower0.7ex\hbox{ $ {\text{3}} $ }}}}{{{{\text{v}}_{\text{1}}}}} \\
\end{gathered} $
Now the acceleration of the particle:
 $ s = ut + \dfrac{1}{2}a{t^2} $
On putting the values,
 $ 12 = 3u + \dfrac{1}{2}a \times {3^2} $
And
 $ 12 = 3u + \dfrac{9}{2}a $
Similarly,
 $ 42 = 6u + \dfrac{1}{2}a \times 36 $
From the above equations,
 $ a = 2m/{s^2} $
 $ u = 1m/s $
Thus, the initial velocity of the particle is 1m/s.
The correct option is E.

Additional Information
Speed is a scalar quantity that refers to "how fast an object is moving." Speed can be thought of as the rate at which an object covers distance. The first scientist to measure speed as distance over time was Galileo. Time in physics is defined by its measurement: time is what a clock reads. In classical, non-relativistic physics, it is a scalar quantity and, like length, mass, and charge, is usually described as a fundamental quantity.

Note
It should always be kept in mind that there is a difference between speed and velocity. Just as distance and displacement have distinctly different meanings, same is the situation between speed and velocity. Velocity is a vector quantity that refers to the rate at which an object changes its position whereas speed is a scalar quantity that refers to how fast an object is moving. Velocity gives us a sense of direction whereas speed does not give any sense of direction.