Collisions and Conservation of Momentum

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Collisions and Conservation of Momentum

Momentum is conserved in a system. See how the conservation of momentum equation is applied to elastic and inelastic collisions.

Learning Targets

I understand that momentum is conserved
I can use the conservation of momentum equation in various situations
I know the difference between elastic and inelastic collisions
I can solve for variables in elastic and inelastic collisions

Name	Variable	MKS Unit	Unit Abbreviation
Momentum	p	kilograms times meters per second	kg∙m/s
Mass	m	kilogram	kg
Velocity	v	meters per second	m/s

Conservation of Momentum

Conservation of Momentum: within a system momentum is conserved

- Reminder: momentum is the product of mass and velocity (p=mv)

Conservation of Momentum Equation

All the momentum in the objects before equals all the momentum in the objects after an interaction

p_1i + p_2i = p_1f + p_2f

m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f

What each variable stands for

m₁: first object mass

v_1i: first object initial velocity

v_1f: first object final velocity

m₂: second object mass

v_2i: second object initial velocity

v_2f: second object final velocity

Conserved Momentum Interactions

Conserved Momentum Interactions Include:

Objects Separating

- Exploding
- Throwing

Starting together with a single velocity:

v_i(m₁ + m₂) = m₁v_1f + m₂v_2f

Here you see a firecracker in one piece moving with a single velocity explode into two pieces each with its own velocity. A pitcher would begin at rest together with a baseball and after both the pitcher and baseball would have a separate velocity. We will pretend that the problems in this section occur on a frictionless surface.

Objects Joining

- Catching
- Some collisions

- Ending together with a single velocity:

m₁v_1i + m₂v_2i= v_f(m₁+ m₂)

Here you see a player catch a ball each with its own velocity and together after the catch with one velocity. In another example you see a dart hit and stick into a dartboard traveling as one after the collision.

Types of Collisions

Elastic Collision
- Momentum and kinetic energy are conserved within the system
- The original objects that collide maintain their form and do not release heat in a perfect elastic collision

Billiard balls colliding is an example of an elastic collision

Elastic Collision Equation
- (objects maintain form and keep separate)

m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f

Inelastic Collision
- Momentum but not kinetic energy is conserved within the system

Inelastic Collision Equation
- (objects combine and stick together)

m₁v_1i + m₂v_2i= v_f(m₁+ m₂)

Q1: In which of these collisions is kinetic energy conserved? (inelastic collision, elastic collision, both)

Elastic Collision

Q2: In which of these collisions is momentum conserved? (inelastic collision, elastic collision, both)

Both Inelastic and Elastic

Example Problems

Q3: A 0.80 kg firecracker is traveling through the air at 12 m/s to the right when it explodes. After the explosion, a 0.30 kg piece of it is flying to the left at 6.0 m/s. What is the mass of the other piece and how fast is it flying?

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Q4: A 95 kg pitcher at rest throws a 0.15 kg baseball 40 m/s to the right. How fast would the pitcher be going after the throw on a frictionless surface?

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Q5: How fast is an 85 kg receiver traveling 6 m/s to the right going after catching a 0.43 kg football traveling at 30 m/s right?

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Q6: A 0.1 kg pool ball traveling 2.5 m/s hits another 0.1 kg at rest. If the first ball stops after the elastic collision, how fast is the second now moving?

(click on picture to enlarge)

Q7: A 0.05 kg dart traveling 16 m/s hits a 0.15 kg movable target and sticks to it. How fast is the dart in the target moving together after the collision?

(click on picture to enlarge)

Conservation of Momentum Problem Set

Check out the quick check answers to see if you are correct. Watch the end of the video lesson at the top of the page to see more about the solution.

1. A 2800 kg truck moving at 12 m/s to the right hits a stopped 1100 kg car. What is the combined velocity the moment they stick together?

8.62 m/s right

2. Joe has a mass of 85 kg and is at rest holding a 0.9 kg snowball. What is Joe’s velocity if he throws the snowball at 13 m/s to the right on a frictionless surface?

0.138 m/s left

3. A 0.45 kg ball moving at 5.1 m/s to the right hits a 0.50 kg ball at rest. If the 0.45 kg ball is moving at 2.0 m/s right after the elastic collision, what velocity is the 0.50 kg balls velocity?

2.79 m/s right

4. A 7.5 kg shopping cart is rolling at 1.0 m/s to the left at the moment you throw a 3 kg bag of flour in it at 5 m/s to the right. What is the combined velocity of the cart with a bag of flour in it?

0.714 m/s right

5. A 50 kg boy is skating down the street on his 1.2 kg skateboard at 6.1 m/s to the right when he jumps off and is then going 4.0 m/s to the right. What is the velocity of the skateboard at this moment?

18.6 m/s right