Question

The initial velocity of a particle is u (at=0) and the acceleration is given by f=at. Which of the following relations is valid?
A) $v = u + a{t^2}$
B) $v = u + \dfrac{{a{t^2}}}{2}$
C) $v = u + at$
D) $v = u$

Answer 1

Hint: Rate of change of velocity is termed as acceleration. When rate of change is added in any definition of it means that the change is taking place with respect to time. By using the above concept we will derive an equation of motion mentioned in the options.

Complete step by step solution:
Let us derive a few terms first like acceleration, velocity (initial and final both).
Initial velocity: initial velocity is the velocity of the object before acceleration.
Final velocity: after acceleration for some amount of time, the new velocity is the final velocity.
Acceleration: the rate of change of velocity of an object. Acceleration is the vector quantity which means it has both magnitude and direction. The orientation of an object’s acceleration is given by the orientation of the net force acting on that object.
Let’s do the derivation of the equation:
Acceleration of the particle is given as f=at in the question.
$ \Rightarrow f = at = \dfrac{{dv}}{{dt}}$ (Acceleration rate of change of velocity)
 Let’s perform the definite integration on both LHS and RHS
$ \Rightarrow \int\limits_u^v {dv = \int\limits_0^t {at(dt)} } $ (We will integrate velocity from initial velocity to final velocity and the term containing t with zero to t as upper and lower limits respectively)
$ \Rightarrow (v - u) = a(\dfrac{{{t^2}}}{2} - \dfrac{0}{2})$ (After integration we have put in the limits)
$
   \Rightarrow v - u = \dfrac{{a{t^2}}}{2} \\
   \Rightarrow v = u + \dfrac{{a{t^2}}}{2} \\
 $ (We got second equation of motion)
Option (B) is correct.

Note: We have numerous examples or the types of motion where the equation of motion is valid or has its applications: translational motion, rotational motion, oscillations etc. Even projectiles do follow the equation of motion be it horizontal or oblique motion.