Question

An impulse is supplied to a moving object with the force at an angle of ${120^ \circ }$ with the velocity vector. The angle between the impulse vector and the change in momentum vector is
A. ${120^ \circ }$
B. ${0^ \circ }$
C. ${60^ \circ }$
D. ${240^ \circ }$

Answer 1

Hint: We know that the impulse is the product of force and the time for which the force acts.
$ \Rightarrow $${\text{impulse}} = F\Delta t$
Where F is a force and $\Delta t$ is the small change in time for which the force acts.
By using Newton's second law we can find that the impulse and the change in momentum are equivalent.

Complete answer:
It is given that an impulse is supplied to a moving object.
We know that impulse is the product of force and the time for which the force acts. In equation form we can write it as
$ \Rightarrow $${\text{impulse}} = F\Delta t$
Where F is a force and $\Delta t$ is the small change in time for which the force acts.
It is given that the force with which impulse is supplied is making an angle of ${120^ \circ }$ with the velocity vector.
This means the impulse vector is making an angle of ${120^ \circ }$ with the direction of velocity because the direction of impulse will be the direction of force.
we need to find the angle made between the impulse vector and the change in momentum vector.
From Newton's second law we know that force is the rate of change of momentum.
That is,
$ \Rightarrow $$F = \dfrac{{\Delta P}}{{\Delta t}}$ ------(1)
Where P is the momentum.
Momentum is the product of mass and velocity of the body. $P = mv$ .
From equation 1 we can get the change in momentum as the product of force and change in time.
That is,
$ \Rightarrow $$\Delta P = F\Delta t$
We can see that the right-hand side of this equation is the same as that of impulse.
Thus, the impulse and change in momentum are the same.
Which means the direction of the impulse vector and the direction of change in momentum vector is the same.
So, they will not make any angle.
The angle between the impulse vector and change in momentum vector will be ${0^ \circ }$.

Hence the correct answer is option B.

Note:
Remember that the impulse is the product of force and the time for which the force acts. Since force is a vector quantity the impulse is also a vector quantity. Since time does not have any direction the direction of the impulse vector will be the same as that of the applied force. The final momentum can have any direction but the vector denoting the change in momentum, will always be in the same direction.