CHAPTER 3 MOMENTUM AND IMPULSE prepared by Yew Sze Ling@Fiona, KML

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Chapter 3 Momentum and Impulse

Curriculum Specification Remarks

Before After Revision

3.1 Momentum and impulse

a) Define momentum and impulse, tFJ

. (C1, C2)

b) Solve problem related to impulse and impulse-

momentum theorem, if vmvmpJ

. (C3, C4)

c) Use F-t graph to determine impulse. (C3, C4)

3.2 Conservation of linear momentum

a) State the principle of conservation of linear momentum

(C1, C2)

b) Apply the principle of conservation of momentum in

elastic and inelastic collisions in 1D and 2D collisions.

(C3, C4)

c) Differentiate elastic and inelastic collisions. (C3, C4)

CHAPTER 3 MOMENTUM AND IMPULSE prepared by Yew Sze Ling@Fiona, KML

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3.1 Momentum and Impulse

Momentum

The linear momentum (or “momentum” for short) of an object is defined as the product of its mass and its velocity.

vmp

SI unit of momentum: -1s m kg or s N

Momentum is vector quantity that has the same direction as the velocity.

Since momentum is a vector quantity, it has direction and can be resolved into x and y-component. Consider a momentum p at an angle θ from the horizontal.

sin

cos

mvmvp

mvmvp

yy

xx

The more momentum the object has, the harder it is to stop it, and the greater effect it will have on another object if it is brought to rest by striking the object.

2000 kg moving at 5 m s-1

0.02 kg moving at 400 m s-1

Both are hard to stop because both have great momentum. Charging elephant has great mass,

single bullet has high velocity.

Impulse

The impulse of a force is the product of the average force and the time interval during which the force acts.

tFJ

SI unit of impulse: -1s m kg or s N

Impulse is a vector quantity whose direction is the same as the constant force on the object

F-t Graph

If the force varies with time as shown in Figure below, impulse is equal to the shaded area under F –t graph.

px

py

θ

Area = F ×Δt

approximate

CHAPTER 3 MOMENTUM AND IMPULSE prepared by Yew Sze Ling@Fiona, KML

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Impulse-Momentum Theorem

When a net average force ∑ ̅⃗ acts on an object during a time interval Δt, the impulse ⃗ of this

force is equal to the change in momentum Δ ⃗⃗⃗ of the object:

pJ

if vmvmtF

Impulse – momentum relationship:

If the change in momentum occurs over a long time, the force of impact is small.

If the change in momentum occurs over a short time, the force of impact is large.

Besides, same impulse does not mean same amount of force or the same amount of time. It means the same product of force and time. For example:

Conclusion

Even though the impulse are the same in both object cases, object with larger time in contact

will experience less damage compare to the object with shorter time in contact.

Egg breaks!

Egg survives!

mu

mu

Impulse on

hammer

Impulse on

nail

CHAPTER 3 MOMENTUM AND IMPULSE prepared by Yew Sze Ling@Fiona, KML

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3.2 Conservation of Linear Momentum

Principle of conservation of linear momentum states that the total linear momentum of an

isolated system remains constant (is conserved).

if pp

22112211 iiff vmvmvmvm

An isolated system is one for which the vector sum of the average external forces acting on the

system is zero.

Collision in One Dimensional Motion

Elastic collision Inelastic collision Perfectly inelastic collision

Total momentum is conserved:

fi pp

22112211 vmvmumum

Total momentum is conserved:

fi pp

22112211 vmvmumum

Total momentum is conserved:

fi pp

vmmumum 212211

Total kinetic energy is conserved:

fi KK

No loss of kinetic energy in

the collision.

Total kinetic energy is NOT conserved:

fi KK

Some of the initial kinetic energy is transformed into another

type of energy such as heat energy due to friction.

The total kinetic energy after the collision is less than the total

kinetic energy before the collision.

However, total energy is conserved.

Caution: It is a common misconception that the only inelastic collisions are those in which the

colliding bodies stick together. In fact inelastic collisions include many situations in which the bodies do not stick.

K.E. is conserved K.E. is NOT conserved

CHAPTER 3 MOMENTUM AND IMPULSE prepared by Yew Sze Ling@Fiona, KML

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Momentum along x-axis:

Momentum along y-axis:

Other example of 1D collision: Explosion

Collision in Two Dimensional Motions

For collision in two dimensions, we MUST conserve the momentum component by component.

To do this we set up a coordinates system and resolve the velocities into x and y component.

Example:

Initially, object m1 is moving with velocity u1 along

the x axis only.

After the collision, two objects move off in different

direction. Thus, we have to resolve the velocities (vector) into x and y component.

Additional Knowledge: Coefficient of restitution(Not in Syllabus)

When there is a head on collision between two bodies, the ratio of their relative velocity after collision and their relative velocity before collision is called the coefficient of restitution. Thus

12

12

uu

vve

For (perfectly) elastic collision, e = 1

If 0 < e < 1, then the collision is inelastic (objects do not stick together after collision)

For perfectly inelastic collision, e = 0 (objects stick together after collision)

Total momentum is conserved:

fi pp

22110 vmvm

Total kinetic energy is NOT conserved:

fi KK

m1 m2

m1

m2

u = 0 m s-1

m1 m2

v1 v2

CHAPTER 3 MOMENTUM AND IMPULSE prepared by Yew Sze Ling@Fiona, KML

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Exercise

Momentum and Impulse

1. A 1.5 kg ball was kicked with initial velocity of 40 m s-1

at the angle 30° with the horizontal

line. Calculate the momentum of the ball and also the horizontal and vertical components of the momentum.

2. A 46-kg skater is standing still in front of a wall. By pushing against the wall she propels

herself backward with a velocity of ‒1.2 m/s. Her hands are in contact with the wall for 0.80 s.

Ignore friction and wind resistance. Find the magnitude and direction of the average force she exerts on the wall

3. A constant friction force of 25 N acts on a 65.0 kg skier for 15 s on level snow. What is the

skier’s change in velocity?

4. A golf ball strikes a hard, smooth floor at an angle of 30.0° and,

as the drawing shows, rebounds at the same angle. The mass of

the ball is 0.047 kg, and its speed is 45 m s-1

just before and after

striking the floor. What is the magnitude of the impulse applied to

the golf ball by the floor?

(Hint: Note that only the vertical component of the ball’s momentum changes during impact with the floor)

5. The graph shows the force acting on a tennis ball of mass 60 g during a return shot.

a. What is the impulse on the ball?

b. If the ball reaches the player with velocity of 22 m s-1

moving to the left, what is the

velocity of the return shot to the right?

Conservation of Linear Momentum

1. A 5.00-kg ball, moving to the right at a velocity of 2.00 m s-1

on a frictionless table, collides

head-on with a stationary 7.50-kg ball. Find the final velocities of the balls if the collision is

a. elastic b. completely inelastic

2. A two-stage rocket moves in space at a constant velocity of 4900 m s-1

. The two stages are then

separated by a small explosive charge placed between them. Immediately after the explosion

the velocity of the 1200 kg upper stage is 5700 m s-1

in the same direction as before the

explosion. What is the velocity (magnitude and direction) of the 2400 kg lower stage after the

explosion?

3. An object A of mass 2.0 kg moves to the right at a speed of 5.0 m s-1

. Object B of mass 4.0 kg

moves to the left at a speed of 8.0 m s-1

. Determine the velocity of A and B after the head-on

collision, assuming that the collision is elastic.

200 N

CHAPTER 3 MOMENTUM AND IMPULSE prepared by Yew Sze Ling@Fiona, KML

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4. By accident, a large plate is dropped vertically and breaks into three

pieces. The pieces fly apart parallel to the floor. Using the data shown in the drawing, find the masses of pieces 1 and 2.

5. The drawing shows a collision between two pucks on an air-

hockey table. Puck A has a mass of 0.025 kg and is moving

along the x-axis with a velocity of +5.5 m s-1

. It makes a

collision with puck B, which has a mass of 0.050 kg and is

initially at rest. The collision is not head-on. After the

collision, the two pucks fly apart with the angles shown in the drawing. Find the final speeds of

a. puck A

b. puck B

6. A mine car (mass = 440 kg) rolls at a speed of 0.50 m s-1

on a

horizontal track, as the drawing shows. A 150 kg chunk of coal

has a speed of 0.80 m s-1

when it leaves the chute. Determine the

speed of the car–coal system after the coal has come to rest in the car.

HOTS Questions

1. A neon atom (m = 20.0 u) makes a perfectly elastic collision with another atom at rest. After

the impact, the neon atom travels away at a 55.6° angle from its original direction and the unknown atom travels away at a angle ‒55.0°. What is the mass (in u) of the unknown atom?

2. A 0.25 kg skeet (clay target) is fired at an angle of 28° to the horizontal with a speed of 25 m s-1

(Figure). When it reaches the maximum height, h, it is hit from below by a 15 g pellet traveling

vertically upward at a speed of 230 m s-1

. The pellet is embedded in the skeet.

a. How much higher, h’, does the skeet go up?

b. How much extra distance, Δx, does the skeet travel because of the collision?