Momentum

A model for collisions

Mr. Porter

2025 - Regents Physics

Momentum is a

vector quantity!

How do you change

Momentum?

With Forces!

Changing Momentum

• Why do baseball players wear gloves to catch baseballs?
• Why do gymnasts practice and perform on mats?
• Why do football players wear padding?
• Why do cars have air bags?
• Why does a boxer "roll with the punch"?
• Why or how are jackhammers able to break concrete?
• Why are tennis players, as well as players in many sports, taught to "follow through" with their swing?

Change in $\vec{p}$ and Forces

• Start with N2L: $\vec{F}_{net} = m\vec{a}$

• Sub in definition of acceleration $a = \frac{\Delta v}{\Delta t}$

• $\vec{F}_{net} = m \frac{\Delta v}{\Delta t} \rightarrow \vec{F}_{net} \Delta t = m \Delta v$

• Rearrange and sub in $\Delta \vec{p}$ for $m \Delta v$

• $\boxed{\vec{F}_{net} \Delta t = \Delta p}$

Impulse

• Impulse is equal to the product of net force and the time interval
• $\Delta \vec{p} = m \Delta \vec{v}$
• Impulse is equal to the change in momentum of an object or system

Water Ballon Toss

What is the best technique and why?

I bet you cannot break

the Egg

Impulse

• Can make impulse changes with large Forces:

$$\huge{F} \tiny{\Delta t} \normalsize = \Delta P$$

• Can make impulse changes with large time intervals:

$$\tiny{F} \huge{\Delta t} \normalsize = \Delta P$$

Momentum Questions - Explain the Following with $J$ and $\vec{p}$:

• Why do baseball players wear gloves to catch baseballs?
• Why do gymnasts practice and perform on mats?
• Why do football players wear padding?
• Why do cars have air bags?
• Why does a boxer "roll with the punch"?
• Why or how are jackhammers able to break concrete?
• Why are tennis players, as well as players in many sports, taught to "follow through" with their swing?

Calculating Momentum Change

Two ways:

• $\Delta p = m \Delta v$
• $\Delta p = F\Delta t$

Momentum Units:

• kg$\cdot$m/s
• N$\cdot$s

Calculating Momentum Change

• We can combine the two equations for momentum change:

$$J = F \Delta t = m \Delta v = m (v-v_0)$$

Impulse & Graphs

How can you calculate the change in momentum from a force vs. time graph?

Changing Force?

How would you do this for a changing force?

Stop and Check

Packet Pages:

What does it mean

for a quantity to be

conserved?

Conservation of

Cake

Packet Pages 7 - 8

Stop after question #7

Momentum Bar Graphs

Momentum is a

conserved quantity

in closed systems

Momentum conservation:

In general momentum of a system is conserved:

$$\boxed{p_{before} = p_{after}}$$

Which can look like...

$$\boxed{m_1\vec{v}_{1i} + m_2\vec{v}_{2i} = m_1\vec{v}_{1f} + m_2\vec{v}_{2f}}$$

Collision Types

• Elastic - Kinetic Energy is conserved, "bouncy"
• Inelastic - Kinetic energy is not conserved, "sticky"
• Explosions - Objects start together and explode away from each other

Elastic Collisions

Inelastic Collisions

Totally Inelastic Sticky Collisions

Explosions

Momentum Bar Graphs

• Y-axis represents velocity (direction matters!)
• x-axis represents mass (width shows size)
• Area of each "bar" represents the momentum

On an icy road, a $5000\textrm{ kg}$ truck rear-ends a $1200 \textrm{ kg}$ car that had been traveling at $13\text{ m/s}$, causing the truck to slow from $14\text{ m/s}$ to $12\text{ m/s}$ and the car to speed up.

1. Sketch the situation
2. Complete the momentum conservation diagram
3. Write momentum conservation equation
4. What was the speed after the collision?

A $6\text{ kg}$ puppy is riding on a skateboard ($1.4\text{ kg}$) moving at a speed of $1.2\text{ m/s}$. While gliding along, the puppy sees a squirrel and jumps off the back of the skateboard with a speed of $0.5\text{ m/s}$ (backwards) relative to the ground. What is the speed of the skateboard just after the puppy jumps? (Be sure to keep careful track of all of the directions involved!)

1. Sketch the situation
2. Complete the momentum conservation diagram
3. Write momentum conservation equation
4. What is the speed of the skateboard just after the puppy jumps?

Two brand new ice skaters have a collision on the ice. Esme, a $21\text{ kg}$ skater, overestimated her new abilities and was gliding across the ice at an ambitious $4.6\text{ m/s}$. Octavia, a $37\text{ kg}$ skater, who was blissfully unaware of Esme’s presence, was gliding in the same direction at a more modest speed of $1.2\text{ m/s}$. When Esme collided with the unsuspecting Octavia, she clung to her for dear life. At what speed did the pair end up traveling at together after the collision?

1. Sketch the situation
2. Complete the momentum conservation diagram
3. Write momentum conservation equation
4. At what speed did the pair end up traveling at together after the collision?

It’s moving day for Keanu’s family. A small dresser is strapped onto a rolling dolly, and Keanu is supposed to push the dresser out to the moving truck. Keanu is annoyed that they have to help, so instead they throw a $300\text{ gram}$ rubber ball at the dresser. The ball is thrown with a speed of $15\text{ m/s}$ and bounces straight back with a speed of $10\text{ m/s}$. Unfortunately, this only results in the dresser rolling at a paltry speed of $0.21\text{ m/s}$, and Keanu eventually decides to just push the dresser anyway. What is the mass of the dresser and dolly combo?

In Case A, a metal bullet penetrates a wooden block. In Case B, a rubber bullet with the same initial speed and mass bounces off of an identical wooden block.

Will the speed of the wooden block after the collision be (i) greater in Case A, (ii) greater in Case B, or (iii) the same in both cases? _____

Explain your reasoning.

In a railroad yard, a train is being assembled. An empty boxcar, coasting at $3\text{ m/s}$, strikes a loaded car that is stationary, and the cars couple together. Each of the boxcars has a mass of $9000\text{ kg}$ when empty, and the loaded car contains $55,000\text{ kg}$ of lumber.

An astronaut of mass $80\text{ kg}$ carries an empty oxygen tank of mass $10\text{ kg}$. By pushing the tank away with a speed of $2.0\text{ m/s}$, the astronaut recoils in the opposite direction. Find the speed with which the astronaut moves off into space.

A $50\text{ kg}$ cart is moving across a frictionless floor at $2.0\text{ m/s}$. A $70\text{ kg}$ boy, riding in the cart, jumps off so that he hits the floor with zero velocity. What was the velocity of the cart after the boy jumped?

A bullet of mass $m$ is moving horizontally with speed $v_0$ when it hits a block of mass $100m$ that is at rest on a horizontal frictionless table. The surface of the table is a height $h$ above the floor. After the impact, the bullet and the block slide off the table and hit the floor a distance $x$ from the edge of the table. Derive expressions for the following quantities in terms of $m$, $h$, $v_0$, and appropriate constants:

1. the speed of the block as it leaves the table
2. the distance x

Suppose that the bullet passes through the block instead of remaining in it.

3. State whether the time required for the block to reach the floor from the edge of the table would now be greater,
   less, or the same. Justify your answer.
4. State whether the distance x for the block would now be greater, less, or the same. Justify your answer.

A track consists of a frictionless arc $XY$, which is a quarter-circle of radius $R$, and a rough horizontal section $YZ$. Block $A$ of mass $M$ is released from rest at point $X$, slides down the curved section of the track, and collides instantaneously and inelastically with identical block $B$ at point $Y$. The two blocks move together to the right, sliding past point $P$, which is a distance $L$ from point $Y$. The coefficient of kinetic friction between the blocks and the horizontal part of the track is $\mu$ Express your answers in terms of $M$, $L$, $\mu$, $R$, and $g$.