Chapter 7

Impulse and Momentum

7.1 The Impulse-Momentum Theorem

There are many situations when the force on an object is not constant.

7.1 The Impulse-Momentum Theorem

DEFINITION OF IMPULSE

The impulse of a force is the product of the averageforce and the time interval during which the force acts:

tFJ

Impulse is a vector quantity and has the same directionas the average force.

s)(N secondsnewton

7.1 The Impulse-Momentum Theorem

tFJ

7.1 The Impulse-Momentum Theorem

DEFINITION OF LINEAR MOMENTUM

The linear momentum of an object is the product of the object’s mass times its velocity:

vpm

Linear momentum is a vector quantity and has the same direction as the velocity.

m/s)(kg ndmeter/secokilogram

7.1 The Impulse-Momentum Theorem

of vvFmmt

final momentum initial momentum

IMPULSE-MOMENTUM THEOREM

When a net force acts on an object, the impulse ofthis force is equal to the change in the momentumof the object

impulse

7.2 The Principle of Conservation of Linear Momentum

Internal forces – Forces that objects withinthe system exert on each other.

External forces – Forces exerted on objectsby agents external to the system.

7.2 The Principle of Conservation of Linear Momentum

PRINCIPLE OF CONSERVATION OF LINEAR MOMENTUM

The total linear momentum of an isolated system is constant(conserved). An isolated system is one for which the sum ofthe average external forces acting on the system is zero.

7.2 The Principle of Conservation of Linear Momentum

PRINCIPLE OF CONSERVATION OF LINEAR MOMENTUM

The total linear momentum of an isolated system is constant(conserved). An isolated system is one for which the sum ofthe average external forces acting on the system is zero.

In the top picture the net external force on thesystem is zero.

In the bottom picture the net external force on thesystem is not zero.

7.3 Collisions in One Dimension

The total linear momentum is conserved when two objectscollide, provided they constitute an isolated system.

Elastic collision -- One in which the total kinetic energy of the system after the collision is equal to the total kinetic energy before the collision.

Inelastic collision -- One in which the total kinetic energy of the system after the collision is not equal to the total kinetic energy before the collision; if the objects stick together after colliding, the collision is said to be completely inelastic.

7.3 Collisions in One Dimension

Example 8 A Ballistic Pendulum

The mass of the block of woodis 2.50-kg and the mass of the bullet is 0.0100-kg. The blockswings to a maximum height of0.650 m above the initial position.

Find the initial speed of the bullet.

7.3 Collisions in One Dimension

22112211 ooff vmvmvmvm

Apply conservation of momentum to the collision:

1121 of vmvmm

1

211 m

vmmv fo

7.3 Collisions in One Dimension

Applying conservation of energyto the swinging motion:

221 mvmgh

2212

121 ff vmmghmm

221

ff vgh

m 650.0sm80.922 2 ff ghv

7.3 Collisions in One Dimension

1

211 m

vmmv fo

m 650.0sm80.92 2fv

sm896m 650.0sm80.92kg 0.0100

kg 50.2kg 0100.0 21

ov

7.4 Collisions in Two Dimensions

xoxoxfxf vmvmvmvm 22112211

yoyoyfyf vmvmvmvm 22112211

7.5 Center of Mass

The center of mass is a point that represents the average location forthe total mass of a system.

21

2211

mm

xmxmxcm

Problems to be solved

• 7.8, 7.34, 7.41, 7.44, 7.60, 7.64

• B7.1 A 1000kg car moving at 20m/s hits a 2000kg parked car head on. How much mechanical energy is dissipated if the collision is (a) elastic, (b) completely inelastic? Ans.: (a)None, because mechanical energy is conserved during elastic collisions. (b) 13.33×104J

• B7.2 A bullet of mass 10-2kg and initial horizontal velocity of 250m/s strikes and is embedded in a 1kg wooden block. The wooden block hangs on the end of a long string. (a) What is the velocity of the block and bullet after impact? (b) How high will the block and bullet swing upward? Ans: (a) 2.48m/s (b) 0.31m

• B7.3 Two cars approaching each other along streets that meet at a right angle collide at the intersection. After the crash, they stick together. If one car has a mass of 1450kg and an initial speed of 11.5m/s and the other has a mass of 1750kg and an initial speed of 15.5m/s, what will be their speed and direction immediately after impact? Ans: 9.95m/s ; 58.4º

• B7.4 A billiard ball moving at 10.0m/s along the positive x-axis collides with a second billiard ball at rest. The balls have identical masses. After the collision, the incoming (or incident) ball moves on with a speed of 7.7m/s at an angle of 40⁰ from the x-axis. What are the speed and direction of motion of the struck ball?

Ans: 6.43m/s ; -50.37º