Impulse - Momentum Theorem

Impluse And Momentum of Class 11

In the previous chapter, we have learnt that work done by a force brings about change in kinetic energy of a particle. Let us see what physical quantity changes due to impulse of a force.

According to Newton's second law, the net force acting on a particle is equal to the product of mass and acceleration.

Impulse - Momentum Theorem

Since a = Impulse - Momentum Theorem, therefore

Impulse - Momentum Theorem= mImpulse - Momentum Theorem

Substituting the value of net force in equation (9.3), we get

Impulse - Momentum Theorem = Impulse - Momentum Theorem

orImpulse - Momentum Theorem= Impulse - Momentum Theorem

Notice when we change the variable of integration from t to v, we must also change the time limits of the integral to the corresponding limits of v.

For constant mass,

Impulse - Momentum Theorem = Impulse - Momentum Theorem

orImpulse - Momentum Theorem= Impulse - Momentum Theorem(9.4)

The quantity

Impulse - Momentum Theorem(9.5)

is a vector and is called the momentum of a particle of mass m moving with velocity Impulse - Momentum Theorem.

ThusImpulse - Momentum Theorem= Impulse - Momentum Theorem

orImpulse - Momentum Theorem= ΔImpulse - Momentum Theorem (9.6)

Equation (9.6) shows that the net impulse of forces acting on a particle is equal to the change in momentum of the particle. This is called the Impulse-Momentum Theorem.

Example 9.3

A 2 kg is moving at a speed of 6 m/s. How large a force F is needed to stop the block in a time of 0.5 ms?

Solution

Impulse on block = Change in momentum of block

Ft = mvf - mvi

F(5 × 10-4) = 2(0) - (2)(6)

orF = -2.4 × 104 N

The negative sign indicates that the force opposes the motion.

Example 9.4

A ball falling with velocity Impulse - Momentum Theoremm/s subjected to a net impulse
Impulse - Momentum Theorem = (0.6 Impulse - Momentum Theorem + 0.18 Impulse - Momentum Theorem) Ns. If the ball has a mass of 275 g, calculate its velocity immediately following the impulse.

Solution

Using Impulse - Momentum Theorem

mImpulse - Momentum Theorem - mImpulse - Momentum Theorem= Impulse - Momentum Theorem

orImpulse - Momentum Theorem

Thus, Impulse - Momentum Theorem = -0.65 Impulse - Momentum Theorem - 0.35 Impulse - Momentum Theorem + 0.6 Impulse - Momentum Theorem+ 0.18 Impulse - Momentum Theorem/0.275

orImpulse - Momentum Theorem = (-0.65 Impulse - Momentum Theorem - 0.35 Impulse - Momentum Theorem) + (2.18 Impulse - Momentum Theorem + 0.655Impulse - Momentum Theorem)

orImpulse - Momentum Theorem = (1.53 Impulse - Momentum Theorem + 0.305 Impulse - Momentum Theorem) m/s

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