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phys: clarify momentum and impulse based on video

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eggy 2020-10-20 21:37:43 -04:00
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@ -404,27 +404,27 @@ Newton's second law of motion states that the **net** force on an object is equa
### Momentum
The momentum of an object describes the difficulty of changing its motion. It is expressed mathematically as $\vec{P}$, and is equal to mass times initial velocity, or:
The momentum of an object describes the difficulty of changing its motion. Its unit is **Newton seconds** ($\text{Ns}$) and it is expressed mathematically as $\vec{P}$ such that $\vec{P}$ is equal to mass times velocity, or:
$$\vec{P} = m\vec{v}$$
Newton's second law can be manipulated so that momentum of an object — if its mass is constant — is equal to the net average force on that object multiplied by time.
$$
\Sigma \vec{F}_\text{avg} = m\vec{a} \\
= m(\frac{\vec{v_2}-\vec{v_1}}{\Delta t}) \\
= \frac{m \vec{v_2} - m \vec{v_1}}{\Delta t}) \\
= \frac{\vec{P_2} - \vec{P_1}}{\Delta t} \\
\Sigma \vec{F}_\text{avg} = \frac{\Delta \vec{P}}{\Delta t} \\
\Delta \vec{P} = \Sigma \vec{F}_\text{avg} · \Delta t
$$
This means that to achieve the *same momentum*, a smaller force can be applied for a longer period of time or a larger force can be applied in a smaller period of time.
### Impulse
Impulse is the average net force $\Sigma F_{\text{avg}}$ applied over a time interval $\Delta t$. It is expressed mathematically as $\vec{J}$, and is represented in **Newton seconds** ($\text{Ns}$). It is equal to:
Impulse is the change in momentum of an object because of a force. It is expressed mathematically as $\vec{J}$, and is also represented in Newton seconds.
$$\vec{J} = \Sigma \vec{F}_\text{avg} · \Delta t$$
This can be rearranged and combined with other formulas, if the mass of the object is **constant**, so that impulse is equal to the difference in linear momentum.
$$
\vec{J} = m\vec{a_\text{avg}} · \Delta t \\
\vec{J} = mt (\frac{\vec{v_2} - \vec{v_1}}{\Delta t}) \\
\vec{J} = m\vec{v_2} - m\vec{v_1} \\
\vec{J} = \vec{P}_2 - \vec{P}_1 \\
\vec{J} = \Delta \vec{P}
$$
Using the above, the net average force on an object can also be written as the rate of change of its momentum with respect to time.
$$
\Sigma \vec{F}_\text{avg} = \frac{\Delta \vec{P}}{\Delta t}
$$
Note that that is also equal to $\Delta \vec{P}$ above.
## 3.1 - Thermal concepts