# Momentum

Momentum is how much something wants to **keep moving** in the same direction.

This truck would be hard to stop ...

... it has a **lot of momentum**.

**Faster?** More momentum! **Heavier?** More momentum!

Momentum is mass times velocity.

The symbol is **p**:

**p** = m **v**

### Example: What is the momentum of a 1500 kg car going at highway speed of **28 m/s** (about 100 km/h or 60 mph)?

**p** = m **v**

**p** = 1500 kg × **28 m/s**

**p** = 42,000 kg m/s

The unit for momentum is:

- kg m/s (kilogram meter per second), or
- N s (Newton second)

They are the same! **1 kg m/s = 1 N s**

We will use both here.

More examples:

Mass | Speed | Momentum | |
---|---|---|---|

Bullet (9 mm) | 7.5 g 0.0075 kg | 1000 m/s | 0.0075 × 1000 = 7.5 kg m/s |

Tennis Ball | 57 g 0.057 kg | 50 m/s | 0.057 × 50 = 2.85 kg m/s |

Soccer Ball | 16 oz 0.45 kg | 100 km/h 28 m/s | 0.45 × 28 = 12.6 kg m/s |

Basket Ball | 22 oz 0.6 kg | 3 m/s | 0.6 × 3 = 1.8 kg m/s |

Hammer | 400 g 0.4 kg | 7 m/s | 0.4 × 7 = 2.8 kg m/s |

Runner | 80 kg | 9 km/h 2.5 m/s | 80 × 2.5 = 200 kg m/s |

Car | 1500 kg | 100 km/h 28 m/s | 1500 × 28 = 42,000 kg m/s |

Momentum has **direction**: the **exact same direction** as the velocity.

But many examples here only use speed (velocity without direction) to keep it simple.

## Impulse

Impulse is change in momentum. Δ is the symbol for "change in", so:

Impulse is Δp

Force can be calculated from the change in momentum over time (called the "time rate of change" of momentum):

F = \frac{Δp}{Δt}

### Example: You are 60 kg and run at 3 m/s into a wall.

The wall stops you in 0.05 s. What is the force?

The wall is then padded and stops you in 0.2 s. What is the force?

First calculate the impulse:

Δp = m v

Δp = 60 kg x 3 m/s

Δp = 180 kg m/s

Stopping in 0.05 s:

F = \frac{Δp}{Δt}

F = \frac{180 kg m/s}{0.05 s} = 3600 N

Stopping in 0.2 s:

F = \frac{Δp}{Δt}

F = \frac{180 kg m/s}{0.2 s} = 900 N

Stopping at a slower rate has much less force!

- And that is why padding works so well
- And also why crash helmets save lives
- And why cars have crumple zones

### Q: Isn't force normally calculated using F = ma ?

A: Well, F = \frac{Δp}{Δt} is the **same thing**, just a different form:

Start with: | F = ma | |

Acceleration is change in velocity v over time t: | F = m\frac{Δv}{Δt} | |

Rearrange to: | F = \frac{Δmv}{Δt} | |

And Δmv is change in momentum: | F = \frac{Δp}{Δt} |

## Impulse From Force

We can rearrange:

F = \frac{Δp}{Δt}

Into:

Δp = F Δt

So we can calculate the Impulse (the change in momentum) from force applied for a period of time.

### Example: A ball is hit with a 300 N force. High speed cameras show the contact lasted for 0.02 s. What was the impulse?

Δp = F Δt

Δp = 300 N × 0.02 s

Δp = 6 N s

## Momentum is Conserved

**Conserved**: the total stays the same (within a closed system).

**Closed System**: nothing transfers in or out, and no external force acts on it.

### In our Universe:

*Note: At an atomic level Mass and Energy can be converted via E=mc ^{2}, but nothing gets lost.*

## Momentum is a Vector

Momentum is a vector: it has size AND direction.

Sometimes we don't mention the direction, but other times it is important!

### One Dimension

A question may have only one dimension, and all we need is positive or negative momentum:

### Two or More Dimensions

Questions can be in two (or more) dimensions like this one:

### Example: A pool ball bounces!

It hits the edge with a velocity of **8 m/s at 50°**, and bounces off at the same speed and reflected angle.

It weighs 0.16 kg. What is the change in momentum?

Let's break the velocity into x and y parts. Before the bounce:

- v
_{x}= 8 × cos(50°)*...going along* - v
_{y}= 8 × sin(50°)*...going up*

After the bounce:

- v
_{x}= 8 × cos(50°)*...going along* - v
_{y}= 8 × −sin(50°)*...going down*

The x-velocity does not change, but the y-velocity changes by:

Δv_{y} = (8+8) × sin(50°)

= 16 × sin(50°)

And the change in momentum is:

**Δp** = m **Δv**

Δp = 0.16 kg × 16 × sin(50°) m/s

Δp = 1.961... kg m/s

## Animation

Play with momentum in this animation.

### Footnote: The formula

**p** = m **v***Momentum is mass times velocity*

is not the full story!

It is a wonderful and useful formula for **normal every day use**, but when we look at the atomic scale things don't actually collide. They interact from a distance through electro-magnetic fields.

And the interaction does not need mass, because light (which has no mass) can have momentum.