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183_notes:collisions [2015/09/21 02:16] – [Momentum is never conserved] caballero | 183_notes:collisions [2021/04/01 01:58] – stumptyl | ||
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+ | Section 3.10 and 3.12 in Matter and Interactions (4th edition) | ||
+ | |||
===== Colliding Objects ===== | ===== Colliding Objects ===== | ||
- | One situation where the concept of a [[183_notes: | + | One situation where the concept of a [[183_notes: |
- | ==== Momentum is never conserved ==== | + | ====== Momentum is never conserved |
- | In real situations that you have observed in your everyday life, the momentum of a system is never conserved. There are always external interactions that act to change the system' | + | In real situations that you have observed in your everyday life, the momentum of a system is never conserved. There are always external interactions that act to change the system' |
$$\Delta \vec{p}_{sys} = \vec{p}_{sys, | $$\Delta \vec{p}_{sys} = \vec{p}_{sys, | ||
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What you will do is consider when the external interactions are small enough or occur over a short enough time where the impulse delivered by the system' | What you will do is consider when the external interactions are small enough or occur over a short enough time where the impulse delivered by the system' | ||
- | === Sometimes, you can approximate that the system' | + | ==== Sometimes, you can approximate that the system' |
[{{ 183_notes: | [{{ 183_notes: | ||
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$$\vec{p}_{sys, | $$\vec{p}_{sys, | ||
- | The momentum of the system before the collision is equal to the momentum of the after the collision. The concept of a multi-particle system greatly simplifies the situation because there' | + | The momentum of the system before the collision is equal to the momentum of the after the collision. The concept of a [[183_notes: |
In the case you have been reading about, you can write down the momentum before and the momentum after the collision. You will read about a slight simpler case next. | In the case you have been reading about, you can write down the momentum before and the momentum after the collision. You will read about a slight simpler case next. | ||
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$$p_{sys, | $$p_{sys, | ||
+ | ===== Momentum Conservation in One Dimension ===== | ||
+ | [{{183_notes: | ||
+ | \\ | ||
- | ==== Momentum Conservation in 1 dimension ==== | ||
- | |||
- | [{{ 183_notes: | ||
To make this more concrete, consider the situation to the left where a single object (A) is moving towards another single object (B). In this situation, A is moving to the right with a known speed ($v_A$) while object B is at rest. After the collision, which occurs over a short time, A and B are stuck together moving at an unknown speed ($v$). | To make this more concrete, consider the situation to the left where a single object (A) is moving towards another single object (B). In this situation, A is moving to the right with a known speed ($v_A$) while object B is at rest. After the collision, which occurs over a short time, A and B are stuck together moving at an unknown speed ($v$). | ||
- | Because the collision occurs over a short time, the momentum of the system of A and B is conserved, so we can determine the speed with which A and B move together after the collision ((They must move at the same speed, | + | Because the collision occurs over a short time, the momentum of the system of A and B is conserved, so we can determine the speed with which A and B move together after the collision ((They must move at the same speed, otherwise they wouldn' |
$$m_A \vec{v}_A + m_b \vec{v}_B = (m_A + m_B)\vec{v}$$ | $$m_A \vec{v}_A + m_b \vec{v}_B = (m_A + m_B)\vec{v}$$ | ||
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This is the speed that the objects have while moving together. Notice that this speed is less than the initial speed of A ($v< | This is the speed that the objects have while moving together. Notice that this speed is less than the initial speed of A ($v< | ||
- | ==== Momentum Conservation in 2 dimensions | + | ===== Momentum Conservation in Two Dimensions ===== |
Two dimensional cases of momentum conservation are common, because often times the interactions (or collisions) occur on a flat plane (i.e., you can neglect the component of the momentum in the vertical direction). In this case, the momentum is conserved in both directions separately: | Two dimensional cases of momentum conservation are common, because often times the interactions (or collisions) occur on a flat plane (i.e., you can neglect the component of the momentum in the vertical direction). In this case, the momentum is conserved in both directions separately: | ||
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$${p}_{sys, | $${p}_{sys, | ||
- | Notice that these can be different scalar quantities (and can be negative, too): the momentum is conserved in each direction. | + | Notice that these can be different scalar quantities (and can be negative, too): __**the momentum is conserved in each direction.**__ |
==== Examples ===== | ==== Examples ===== | ||
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* [[: | * [[: | ||
* [[: | * [[: | ||
+ | * [[: |