Question

The momentum p (in kg m/s) of a particle is varying with time t (in s) as $p=2+3{{t}^{2}}$. The force acting on the particle at t =3s will be
A. 18N
B.54N
C. 9N
D. 15N

Answer 1

Hint: As a first step, you could recall Newton’s second law in terms of momentum. Thus, you will find the force to be the time rate of change of momentum to be proportional to force. You could differentiate the given variation of linear momentum with time with respect to time and then substitute the value of time at which the force is to be found.
Formula used:
Force,
$F=\dfrac{dp}{dt}$

Complete step by step solution:
In the question we are given the expression of variation of momentum p (in kg m/s) of some particle with time t (in seconds) and we are asked to find the force acting on this particle at time t = 3s.
In order to solve this question, let us recall Newton’s second law of motion. The law states that the time rate of change of momentum of a body is directly proportional to the force applied, which could be mathematically expressed as,
$F=\dfrac{dp}{dt}$ …………………………………. (1)
But we are given the variation of momentum with time as,
$p=2+3{{t}^{2}}$
Differentiating both sides with respect to time, we get,
$\dfrac{dp}{dt}=6t$
From (1) we know that the force here will be given by,
$F(t)=6t$
We are asked to find the force at t = 3s, so,
$F\left( t=3 \right)=6\times 3$
$\Rightarrow F\left( t=3 \right)=18N$
Therefore, we found that the force acting on the particle at t =3s will be 18N.

Hence option A is the correct answer.

Note:
Other than the statement used in the solution, we have another statement for Newton’s second law of motion in terms of acceleration. For constant mass system, by constant factor rule differentiation,
$F=m\dfrac{dv}{dt}$
But we know that the rate of change of velocity is acceleration. Therefore,
$F=ma$
Hence, we see that the net force is directly proportional to acceleration and also acceleration of a body indicates the presence of force on it.