Импульс
Введено| English | Значение |
|---|---|
| momentum/məʊˈmentəm/ | импульс |
| collision/kəˈlɪʒn/ | столкновение |
| conservation of momentum/ˌkɒnsəˈveɪʃn ɒv məʊˈmentəm/ | закон сохранения импульса |
Hard to stop
- An empty shopping trolley is easy to stop. A loaded truck at the same speed is not.
- The truck has much more momentum 动量.
- Momentum links mass, velocity, force and collisions 碰撞 — and it is always conserved.
Momentum
- Momentum is mass times velocity:
- It is a vector — it has a direction (the direction of motion).
- The unit is the kilogram metre per second ($\text{kg}\,\text{m/s}$).
The water an irregular solid pushes up gives its volume
Momentum is a vector.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
Momentum is mass × velocity, and velocity has a direction, so momentum is a vector.
A $1000\ \text{kg}$ car moves at $20\ \text{m/s}$. What is its momentum, in kg m/s?
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
$p = mv = 1000 \times 20 = 20\,000\ \text{kg}\,\text{m/s}$.
Force changes momentum
- A resultant force changes momentum. The force is the change in momentum per second:
- So a given change in momentum over a longer time needs a smaller force.
- This is why crumple zones, airbags and crash mats help: they make the stop take longer, so the force is gentler.
Why does a car's crumple zone reduce the force on the passengers in a crash?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
Since $F = \dfrac{\Delta p}{\Delta t}$, spreading the same momentum change over a longer time gives a smaller force.
A ball's momentum changes by $20\ \text{kg}\,\text{m/s}$ during a push lasting $2.0\ \text{s}$. What is the average force, in N?
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
$F = \dfrac{\Delta p}{\Delta t} = \dfrac{20}{2.0} = 10\ \text{N}$.
Conservation of momentum 动量守恒
- In a collision with no outside force, the total momentum before = total momentum after:
- Here $u$ is a velocity before and $v$ a velocity after.
- Momentum lost by one object is gained by the other.
See momentum being conserved
Two trolleys collide and stick together.
A $2.0\ \text{kg}$ trolley moving at $3.0\ \text{m/s}$ hits a stationary $1.0\ \text{kg}$ trolley and they stick together. What is their speed afterwards, in m/s?
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Before: $2.0 \times 3.0 + 1.0 \times 0 = 6.0\ \text{kg}\,\text{m/s}$. After: $(2.0 + 1.0)v = 6.0$, so $v = 2.0\ \text{m/s}$.
Worked examples
1. A $2.0\ \text{kg}$ trolley moving at $3.0\ \text{m/s}$ hits a stationary $1.0\ \text{kg}$ trolley and they stick. Momentum before $= 2.0 \times 3.0 = 6.0\ \text{kg m/s}$. After, $3.0\ \text{kg} \times v = 6.0$, so $v = 2.0\ \text{m/s}$.
2. A $0.40\ \text{kg}$ ball is kicked from rest to $15\ \text{m/s}$; the foot touches it for $0.050\ \text{s}$. $\Delta p = 0.40 \times 15 = 6.0\ \text{kg m/s}$, so $F = \dfrac{\Delta p}{\Delta t} = \dfrac{6.0}{0.050} = 120\ \text{N}$.
A 4 kg trolley moves at 7 metres per second and is stopped in 0.50 s. What are its momentum and the average stopping force?
4 times 7 is 28. 28 divided by 0.50 is 56 newtons.
What is the momentum of the trolley?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
4 times 7 is 28 kilogram metres per second.
You've got it
- momentum $p = mv$ — a vector, unit $\text{kg}\,\text{m/s}$
- force $F = \dfrac{\Delta p}{\Delta t}$: a longer stop means a smaller force
- conservation: total momentum before = total momentum after a collision
Write from a momentum of 28 kilogram metres per second
- A 4 kg trolley moves at 7 metres per second and then stops in 0.50 s.
- Find the momentum and the average stopping force.