Question

How does acceleration affect momentum?

Answer 1

Hint: In order to solve this question, we have to define the quantity momentum. The direct application of the mathematical definition of the quantity momentum in order to establish its relationship with the quantity acceleration.

Complete answer:
When a body is moving at a velocity, the effect that is produced by the body when it is stopped or when it interacts with another body is not just, defined by the magnitude of the velocity of the body, but also, the combined effect of the mass and velocity of the body, which is equal to another quantity called momentum.
The momentum is mathematically, defined as the product of the mass and velocity of the body.
$p = mv$
where m = mass and v = velocity.
It is a vector quantity whose direction is the same as that of the velocity.
The momentum represents the impact that is produced by the body by the virtue of its mass and its velocity, when the body collides with another body or the change of which, causes an effect known as the force, which is defined as the rate of change of momentum of the body.
The acceleration is defined as the rate of change of velocity with time.
$\Rightarrow a = \dfrac{v}{t}$
where v = velocity and t = time.
Now, let us understand what happens if there is a change in momentum. When momentum changes with respect to time, it is given by –
$\Rightarrow \dfrac{{dp}}{{dt}} = \dfrac{d}{{dt}}\left( {mv} \right)$
When there is change in momentum in macroscopic bodies, the mass does not change but there is change in the velocity. Hence –
$\Rightarrow \dfrac{d}{{dt}}\left( {mv} \right) = m\dfrac{{dv}}{{dt}}$
From the above relation, we have that the rate of velocity per time being equal to acceleration. Thus,
$\Rightarrow m\dfrac{{dv}}{{dt}} = ma$
Thus, rate of change of momentum is directly proportional to acceleration and equal to the mass times acceleration.

Note: The quantity of rate of change of momentum is equal to a quantity known as force, as devised by Newton's Second law of motion. This gives us the famous expression for force, which is –
$F = \dfrac{{dp}}{{dt}} = m\dfrac{{dv}}{{dt}} = ma$