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;
\[F = \dfrac{{dp}}{{dt}}\] where \[F\] is the force acting on a body, \[p\] is the momentum of a body, and \[t\] is time, and \[\dfrac{{dp}}{{dt}}\] 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
\[F = \dfrac{{dp}}{{dt}}\] where \[F\] is the force acting on a body, \[p\] is the momentum of a body, and \[t\] is time, and \[\dfrac{{dp}}{{dt}}\] 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
\[Ft = {p_f} - {p_o}\]
\[ \Rightarrow Ft = \Delta p\]

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,
\[F = \dfrac{{mv - mu}}{t}\]
Hence, simply by cross multiplying we have
\[Ft = mv - mu\]
Hence, we have that
\[I = mv - mu\] which is the impulse-momentum theorem.