# Momentum

## An object in motion possesses energy that is transferred into objects it collides with.

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Momentum (Accelerated)

### What Is Momentum?

Momentum is a property of a moving object that makes it hard to stop. The more mass it has or the faster it’s moving, the greater its momentum. Momentum equals mass times velocity and is represented by the equation:

Momentum = Mass x Velocity

You can see an animation demonstrating the role of mass and velocity in the momentum of moving objects at this URL:

Objective:

Students will learn what momentum is and how to calculate momentum of objects. In addition, students will learn how to use conservation of momentum to solve basic problems.

### Key Equations

p=mvMomentum is equal to the objects mass multiplied by its velocity\begin{align*} p = m v \; \; \text{Momentum is equal to the objects mass multiplied by its velocity}\end{align*}

pinitial=pfinalThe total momentum does not change in closed systems\begin{align*} \sum p_{\text{initial}} = \sum p_\text{final} \; \; \text{The total momentum does not change in closed systems}\end{align*}

Guidance
• Momentum is a vector that points in the direction of the velocity vector. The magnitude of this vector is the product of mass and speed.
• The total momentum of the universe is always the same and is equal to zero. The total momentum of an isolated system never changes.
• Momentum can be transferred from one body to another. In an isolated system in which momentum is transferred internally, the total initial momentum is the same as the total final momentum.
• Momentum conservation is especially important in collisions, where the total momentum just before the collision is the same as the total momentum after the collision.

### Calculating Momentum

#### Example 1

To calculate momentum with the 1st equation above, mass is measured in (kg), and velocity is measured in meters per second (m/s). For example, Cody and his skateboard have a combined mass of 40 kg. If Cody is traveling at a velocity of 1.1 m/s by the time he reaches the bottom of the ramp, then his momentum is:

Momentum = 40 kg x 1.1 m/s = 44 kg ∙ m/s

Note that the SI unit for momentum is kg ∙ m/s.

Q: The combined mass of Jerod and his skateboard is 68 kg. If Jerod goes down the ramp at the same velocity as Cody, what is his momentum at the bottom of the ramp?

A: His momentum is:

Momentum = 68 kg x 1.1 m/s = 75 kg ∙ m/s

#### Example 2

A truck with mass 500 kg and originally carrying 200 kg of dirt is rolling forward with the transmission in neutral and shooting out the dirt backwards. If the truck is originally moving at 2 m/s, how fast will it be moving after it has shot out all the dirt. You may ignore the effects of friction.

##### Solution

To solve this problem we will apply conservation of momentum to the truck when it is full of dirt and when it has dumped all the dirt.

mivi(mt+md)vivfvfvf=mfvf=mtvf=(mt+md)vimt=(500kg+200kg)2m/s500kg=2.8m/sstart by setting the initial momentum equal to the final momentumsubstitute the mass of the truck plus the mass of the dirt in the truck at the initial and final statessolve for the final velocityplug in the numerical values\begin{align*} m_iv_i&=m_fv_f && \text{start by setting the initial momentum equal to the final momentum}\\ (m_t+m_d)v_i&=m_tv_f && \text{substitute the mass of the truck plus the mass of the dirt in the truck at the initial and final states}\\ v_f&=\frac{(m_t+m_d)v_i}{m_t} && \text{solve for the final velocity}\\ v_f&=\frac{(500\;\text{kg} + 200\;\text{kg})*2\;\text{m/s}}{500\;\text{kg}} && \text{plug in the numerical values}\\ v_f&=2.8\;\text{m/s}\\ \end{align*}

#### Example 3

John and Bob are standing at rest in middle of a frozen lake so there is no friction between their feet and the ice. Both of them want to get to shore so they simultaneously push off each other in opposite directions. If John's mass is 50 kg and Bob's mass is 40 kg and John moving at 5 m/s after pushing off Bob, how fast is Bob moving?

##### Solution

For this problem, we will apply conservation of momentum to the whole system that includes both John and Bob. Since both of them are at rest to start, we know that the total momentum of the whole system must always be zero. Therefore, we know that the sum of John's and Bob's momentum after they push off each other is also zero. We can use this to solve for Bob's velocity.

0mbvbvbvbvb=mjvj+mbvb=mjvj=mjvjmb=50kg5m/s40kg=6.25m/s\begin{align*} 0&=m_jv_j+m_bv_b\\ -m_bv_b&=m_jv_j\\ v_b&=-\frac{m_jv_j}{m_b}\\ v_b&=-\frac{50\;\text{kg}*5\;\text{m/s}}{40\;\text{kg}}\\ v_b&=-6.25\;\text{m/s}\\ \end{align*}

The answer is negative because Bob is traveling in the opposite direction to John.

### Practice

At the following URL, review how to calculate momentum, and then solve the problems at the bottom of the Web page.

### Review Questions

1. Which skateboarder has greater momentum?
1. Skateboarder A: mass = 60 kg; velocity = 1.5 m/s
2. Skateboarder B: mass = 50 kg; velocity = 2.0 m/s
2. 5.00kg\begin{align*}5.00\;\mathrm{kg}\end{align*} firecracker explodes into two parts: one part has a mass of 3.00kg\begin{align*}3.00 \;\mathrm{kg}\end{align*} and moves at a velocity of 25.0m/s\begin{align*}25.0 \;\mathrm{m/s}\end{align*} towards the west. The other part has a mass of 2.00kg\begin{align*}2.00 \;\mathrm{kg}\end{align*}. What is the velocity of the second piece as a result of the explosion?
3. A firecracker lying on the ground explodes, breaking into two pieces. One piece has twice the mass of the other. What is the ratio of their speeds?
4. While driving in your pickup truck down Highway 280\begin{align*}280\end{align*} between San Francisco and Palo Alto, an asteroid lands in your truck bed! Despite its 220kg\begin{align*}220 \;\mathrm{kg}\end{align*} mass, the asteroid does not destroy your 1200kg\begin{align*}1200\;\mathrm{kg}\end{align*} truck. In fact, it landed perfectly vertically. Before the asteroid hit, you were going 25m/s\begin{align*}25 \;\mathrm{m/s}\end{align*}. After it hit, how fast were you going?

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