Question

The initial velocity of a particle is u (at t=0) and the acceleration a is given by ft. Which of the following relations is valid ?
a) $ v=u$
b) $ v =u + ft$
c) $v = u + ft^2$
d) $v = u + \dfrac{ft^2}{2}$

Answer 1

Hint: The above relations of the given problems are the equations of motion. Motion of an object or body is defined by three equations of motion which can be proved both mathematically and graphically.
Equation of motion involves initial and final velocities and acceleration.

Complete step by step answer:
First equation of motion.
Acceleration of a body is determined by difference between its final velocity and initial velocity upon time.
$ \Rightarrow a = \dfrac{{v - u}}{t}$ (rate of change of velocity)
$ \Rightarrow at = v - u$
$ \Rightarrow v = u + at$ (first equation proved)
Second equation of motion average velocity is given as;
$ \Rightarrow {v_{avg}} = \dfrac{{v + u}}{2}$
Displacement is equal to;
$s = {v_{avg}} \times t$ ....................1
On substituting the value of average velocity in equation 1,
$s = \dfrac{{v + u}}{2} \times t$ (substituting the value first equation of motion)
$ \Rightarrow 2s = ((u + at) + u) \times t$ (multiply t inside the bracket)
$ \Rightarrow s = \dfrac{{2ut + a{t^2}}}{2}$
$ \Rightarrow s = ut + \dfrac{{a{t^2}}}{2}$ (Second equation of motion)
From the above two proved equations will help us in finding the correct options from the given question;
We do not have any option which matches with the second equation of the motion.
But we have the first equation of motion in the options.
As the acceleration is the variable quantity, in the question a = ft
Thus, we can write as,
$ \Rightarrow \dfrac{{dv}}{{dt}} = ft$
$ \Rightarrow dv = ft.dt$
$ \Rightarrow \int\limits_u^v {dv} = \int {ft.dt} $ (Integrating on both sides)
$ \Rightarrow v - u = f\dfrac{{{t^2}}}{2}$
$ \Rightarrow v = u + f\dfrac{{{t^2}}}{2}$ (We obtain this equation which is similar to first equation of motion)

Hence, the correct answer is option (D).

Note: Equation of motion has many applications such as in finding out the value of variables in problems of position dependent conservative forces, linear restoring forces, constant force problem, in calculating the constrained motion, velocity dependent forces and for the systems with variable mass.