Question

A body, whose momentum is constant, must have constant:
(A) Force
(B) Velocity
(C) Acceleration
(D) All of these

Answer 1

Hint: - Momentum is simply defined as the mass in motion. All the objects have mass, consequently, if the object is moving it possesses momentum. The momentum of a body is the product of its mass and the velocity by which the object is moving. The point to be kept in mind while solving this question is that the momentum is directly proportional to its velocity, keeping its mass to be constant.

Complete step-by-step solution:
In Newtonian mechanics, momentum is the product of the mass and velocity of a body. If $m$ is the mass of an object and $v$ be its velocity, then the body’s momentum $p$ is:
$p = mv$ ........... $\left( 1 \right)$
Since the momentum is directly proportional to its velocity. If the momentum of a body is constant then its velocity also becomes constant.
As we know that the rate of change in velocity is the acceleration,
$a = \dfrac{{dv}}{{dt}}$ ............. $\left( 2 \right)$
Since the velocity becomes constant,
$\therefore dv = 0$
Substitute the value in the equation $\left( 2 \right)$ ;
$ \Rightarrow a = \dfrac{0}{{dt}}$
$ \Rightarrow a = 0$
So, the acceleration of the body having momentum constant is zero.
Thus we can say that when the momentum of a body is kept constant then its acceleration becomes zero or constant.
Furthermore, Newton’s second law of motion states that the rate of change of momentum of a body is equal to the net force acting on it. If $\Delta p$ be the change in momentum and $\Delta t$ be the time interval, then the net force acting on the body can be given:-
$F = \dfrac{{\Delta p}}{{\Delta t}}$ ............... $\left( 3 \right)$
We have given that the momentum of a body is constant
$\therefore \Delta p = 0$
Substitute the value in the equation $\left( 3 \right)$
$F = \dfrac{0}{{\Delta t}}$
$ \Rightarrow F = 0$
Thus the net force applied on a body having a momentum constant is zero.

As a result, the right option is (D) All of these.

Note:
It is important to note that for a closed system, the total momentum is constant. This is well-known as the ‘law of conservation of momentum’. This law applies to all interactions, including collisions, no matter how complicated the force is between the particles.