Question

State and prove the impulse-momentum theorem.

Answer 1

Hint: Impulse can be defined mathematically as the product of force and time. The Impulse momentum theorem can be gotten from Newton’s second law.
Formula used: In this solution we will be using the following formulae;
$$F={\displaystyle \frac{dp}{dt}}$$ where $$F$$ is the force acting on a body, $$p$$ is the momentum of a body, and $$t$$ is time, and $${\displaystyle \frac{dp}{dt}}$$ signifies instantaneous rate of change of momentum.

Complete Step-by-Step solution:
Generally, impulse is defined as the product of and time. It is generally used to quantify how long a force acts on a particular body. Its unit in Ns. 1 Ns is defined as the amount of impulse when 1 N of force acts on a body for one second. However, by relation, it is equal to the change in momentum of the body
The impulse – momentum theorem generally states that the impulse applied to a body is equal to the change in momentum of that body. This theorem can be proven from Newton’s law.
According to Newton’s second law, we have that
$$F={\displaystyle \frac{dp}{dt}}$$ where $$F$$ is the force acting on a body, $$p$$ is the momentum of a body, and $$t$$ is time, and $${\displaystyle \frac{dp}{dt}}$$ signifies instantaneous rate of change of momentum.
Hence, by cross multiplying, we have
$$Fdt=dp$$
Then, integrating both sides from initial point to final point for both momentum and time, we have
$${\int }_0^{1}Fdt={\int }_{p_0}^{P_f}dp$$
Hence, by integrating, we have that
$$Ft=p_f-p_o$$
$$\Rightarrow Ft=\mathrm{\Delta }p$$

Hence, the impulse is equal to change in momentum.

Note: Alternately, we can use the constant form of Newton's second law equation. Which can be given as,
$$F={\displaystyle \frac{mv-mu}{t}}$$
Hence, simply by cross multiplying we have
$$Ft=mv-mu$$
Hence, we have that
$$I=mv-mu$$ which is the impulse-momentum theorem.