Question

What is the impulse of a body is equal to:
(A) Rate of change in momentum
(B) Change in momentum
(C) The product of the force applied on it and the time of application of force
(D) Both (B) and (C)

Answer 1

We need to derive the equation of force from Newton’s second law of motion. Then on relating this to the definition of impulse we will get the correct answer from the options given in the question.

Formula Used: The following formulas are used to solve this question.
$F = \dfrac{{dp}}{{dt}}$ where $p$ is the linear momentum and $t$ is the change in time.
$J = \int {F \cdot t}$ where $J$ is the impulse, resultant force is $F$ and time is $t$.
$J = \int\limits_{{t_1}}^{{t_2}} {\dfrac{{dp}}{{dt}}} dt$ where impulse is integral over the closed interval ${t_2}$ to ${t_1}$ where ${t_2}$ is the final time and ${t_1}$ is the initial time.

Complete step by step answer
Impulse (symbolized by $J$ or Imp) is the integral of a force, $F$, over the time interval, $t$, for which it acts.
Since force is a vector quantity, impulse is also a vector quantity.
Impulse applied to an object produces an equivalent vector change in its linear momentum, also in the same direction. The SI unit of impulse is the Newton second ( $N \cdot s$ ).
The impulse ( $J$ ) is the integral of the resultant force (F) with respect to time ( $t$ ):
$J = \int {F \cdot t}$ where $J$ is the impulse, resultant force is $F$ and time is $t$.
Thus, impulse of a body is equal to the product of the force applied on it and the time of application of force. Thus, option C is true.
From Newton’s second law, force $F$ is the rate of change of momentum.
$F = \dfrac{{dp}}{{dt}}$ where $p$ is the linear momentum and $t$ is the change in time.
According to the impulse-momentum theorem,
$J = \int\limits_{{t_1}}^{{t_2}} {\dfrac{{dp}}{{dt}}} dt$ where impulse is integral over the closed interval ${t_2}$ to ${t_1}$ where ${t_2}$ is the final time and ${t_1}$ is the initial time.
$J = \int\limits_{{p_1}}^{{p_2}} {dp}$ where impulse is integral over the closed interval ${p_2}$ to ${p_1}$ where ${p_2}$ is the final momentum and ${p_1}$ is the initial momentum.
$= {p_2} - {p_1} = \Delta p$ is the change in linear momentum in a time interval.
Thus the impulse of a body is equal to change in linear momentum in a time interval. Thus Option B is true.

$\therefore$ The correct answer is Option D.

Note
In a collision, objects experience an impulse; the impulse causes and is equal to the change in momentum. The result of the force acting for the given amount of time is that the object's mass either speeds up or slows down (or changes direction).