Conservation of Momentum

Unit 6: Momentum and Impulse

Conservation of Momentum

The law of conservation of momentum states that the momentum of a system is constant if no external forces (like friction) are acting on the system

Conservation of Momentum

Example: If you throw a rock forward from a skateboard, you will move backward in response

What it means…

The total momentum of a system will stay the same before and after a collision

Why does Mr. Stickfigure move backwards on the ice?

Before Collision

After Collision

Momentum = zero

(not moving)

p_total = 0 kg∙m/s

Momentum = zero

p_total = 5 + -5

p_total = 0 kg∙m/s

+p

-p

p = -5 kg∙m/s

p = 5 kg∙m/s

Why does Mr. Stickfigure move backwards?

Answer:

• Mr. Stickfigure moves backwards because he threw the ball forward

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Explain the Relationship

• The Law of the Conservation of Momentum states that the total momentum of a system must stay the same before and after
• Before the ball and Mr. Stickfigure had a total momentum of zero, so after the total momentum needed to stay zero

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Support with Data/Observations

• When the ball moved forward it had a positive momentum, so Mr. Stickfigure needed a negative momentum to cancel it out

Why do both skaters move after?

Before

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p_total = 0 kg∙m/s

(not moving)

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After

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p_total = 0 kg∙m/s

Answer the Question

• Both skaters move after in order to conserve momentum

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Explain the Relationship

• The Law of the Conservation of Momentum states that the total momentum of a system must stay the same before and after
• Before the two skaters had a total momentum of zero, so after the total momentum needed to stay zero

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Support with Data/Observations

• When they push against each other one moved forward and had a positive momentum, so the other needed to move backwards to have a negative momentum, so the total momentum stayed zero

Why do both skaters move after?

Recoil

• Recoil is a term that refers to moment when a gun moves backwards after it is shot
• Recoil happens because everything must follow “The Law of the Conservation of Momentum”!!!

Why a gun “recoils”

The total momentum before was zero

So the total momentum after has to be zero

The gun moves with a negative momentum because the bullet moves with a positive momentum and they cancel out, the total momentum stays zero

3 Types of Collisions

• Elastic
• Inelastic
• Perfectly Inelastic

Elastic or Inelastic?

An elastic collision loses no energy – the deformation on collision is fully restored

In an inelastic collision, energy is lost and the deformation may be permanent

Elastic

• A collision in which two objects move separately with different velocities, but not permanent deformation

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Inelastic

• A collision in which two objects deform so that the objects move in the same direction but with different final velocities after colliding

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Perfectly Inelastic

• A collision in which two objects stick together and move with the same velocity after colliding

For each of the following examples, identify the type of collision…

Perfectly Inelastic

Perfectly Inelastic

Elastic

Perfectly Inelastic

Inelastic

Perfectly Inelastic

Elastic

Explain why the final velocity of the moving object “makes sense” in order to conserve total momentum.

• After the collision, the 1^st ball transferred its momentum to the 2^nd ball
• Since the balls have the same mass, the velocity of the second ball should be the same as the first in order to conserve momentum

Explain why the final velocity of the moving object “makes sense” in order to conserve total momentum.

• After the collision, the 1^st block transferred its momentum to the 2^nd block
• Since the 2^nd block has more mass, the velocity should be less, in order to conserve momentum

The block is 2x bigger, it is ½ as fast

Explain why the final velocity of the moving object “makes sense” in order to conserve total momentum.

• After the collision, the cart and McDonald are moving
• Since there is more mass, there should be less velocity to conserve momentum

Example 1

A 50 kg girl jumps into a 100 kg raft at rest on the water. If the velocity of the girl is 4 m/s as she jumps, what is the final velocity of the girl and the raft?

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G

R

G

R

m_G = 50 kg

v_G = 4 m/s

m_R = 100 kg

v_R = 0 m/s

m_Gv_G + m_Rv_R = (m_G + m_R)v_f

200 = (150)v_f

v_f = 1.33 m/s

Draw a Picture

Before After

m_total = 150 kg

v_f = ?

(50)(4) + (100)(0) = (50 + 100)(v_f)

Example 2

A 63 kg astronaut throws a 5 kg hammer in a direction away from the shuttle with a speed of 18 m/s, pushing the astronaut back to the shuttle. Assuming that the astronaut and hammer start from rest, find the final speed of the astronaut after throwing the hammer.

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Draw a Picture

Before After

A

H

A

H

m_A = 63 kg

v_A = 0 m/s

m_H = 5 kg

v_H = 0 m/s

m_A = 63 kg

v_A = ?

m_H = 5 kg

v_H = 18 m/s

m_Av_A + m_Hv_H = m_Av_Af + m_Hv_Hf

(63)(0)+(5)(0) = (63)v_A + (5)(18)

0 = (63)v_Af + 90

-90 = (63)v_Af

v_Af = -1.43 m/s

+90 kg∙m/s

-90 kg∙m/s

Example 3

1^st Draw a Picture

Before After

1

2

1

2

m₁ = 15 kg

v₁ = 4 m/s

m₂ = 6.5 kg

v₂ = -2 m/s

m₁ = 15 kg

v_f = ?

m₂ = 6.5 kg

m₁v₁ + m₂v₂ = (m₁ + m₂)v_f

(15)(4)+(6.5)(-2) = (15 + 6.5)(v_f)

47 = 21.5v_A

v_f = 2.19m/s

A 15 kg cart moving to the right with a speed of 4 m/s collides with a 6.5 kg cart moving to the left with a speed of 2 m/s. If the carts stick together, find the final speed of the two carts.

60 -13 = 21.5v_f

Key Concepts

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Elastic

m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f

Inelastic

m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f

Perfectly Inelastic

m₁v_1i + m₂v_2i = (m₁+ m₂)v_f

“Starts from rest” V_i = 0m/s

“Stops” V_f = 0m/s

Lesson Check 6.3