8.7: Summary
In this chapter the concepts of impulse and momentum have been introduced. An impulse is the time integral of a force vector and is related to the momentum change of a point mass during a certain time interval. The change in momentum equals the impulse generated by external forces, this also holds for systems of many point masses. If the impulse, or sum of external forces is zero, the momentum of the system does not change and the law of momentum conservation applies. These principles can be applied to determine velocity changes e.g. during collisions. We summarise the most important concepts and equations from this chapter:
- Impulse and momentum
- Impulse vector: \(\overrightarrow{\boldsymbol{J}}_{12}=\sum_{j} \int_{t_{1}}^{t_{2}} \overrightarrow{\boldsymbol{F}}_{i j} \mathrm{~d} t\)
- Momentum vector: \(\overrightarrow{\boldsymbol{p}}_{i}=m_{i} \overrightarrow{\boldsymbol{v}}_{i}\)
- Principle of impulse and momentum: \(\overrightarrow{\boldsymbol{p}}_{i}\left(t_{1}\right)+\overrightarrow{\boldsymbol{J}}_{12}=\overrightarrow{\boldsymbol{p}}_{i}\left(t_{2}\right)\)
- Important assumptions during impulse: 1. Force is very high and 2. time duration of impulse is very short. This allows assuming: 1 . point mass is not moving during impulse. 2. impulse of other forces can be neglected during impulse.
- Use segmented motion to analyse momentum and impulse.
- Impulse and momentum of a system of point masses
- Impulse of CoM: \(\overrightarrow{\boldsymbol{p}}_{G}=m_{\mathrm{tot}} \overrightarrow{\boldsymbol{v}}_{G}=\sum_{i} \overrightarrow{\boldsymbol{p}}_{i}\)
- The sum of all internal forces in a system is zero: \(\sum_{i, j \neq i} \overrightarrow{\boldsymbol{F}}_{i j, \text { int }}=\overrightarrow{\mathbf{0}}\)
- Principle of impulse and momentum for a system:
\[\overrightarrow{\boldsymbol{p}}_{G}\left(t_{1}\right)+\sum_{i} \overrightarrow{\boldsymbol{J}}_{i, \text { ext }, 12}=\overrightarrow{\boldsymbol{p}}_{G}\left(t_{2}\right)\]
- Euler’s first law: \(\sum_{i} \overrightarrow{\boldsymbol{F}}_{i, \text { ext }}=m_{\text {tot }} \overrightarrow{\boldsymbol{a}}_{G}\)
- Momentum conservation
- Law of conservation of momentum: If \(\overrightarrow{\boldsymbol{J}}_{i, \mathrm{ext}, 12}=\overrightarrow{\mathbf{0}}\), then \(\overrightarrow{\boldsymbol{p}}_{G}\left(t_{1}\right)=\) \(\vec{p}_{G}\left(t_{2}\right)\).
- And: \(\overrightarrow{\boldsymbol{v}}_{G}=\) constant.
- Collisions
- Plane of contact, line of impact, draw CS.
- Method for analysing collisions: determine \(\overrightarrow{\boldsymbol{v}}_{G}\), transform to the CoM-frame, \(v_{x 2}^{\prime}=-e v_{x 0}^{\prime}\) and \(v_{y 2}^{\prime}=v_{y 0}^{\prime}\), transform back to original system.
- Coefficient of restitution \(e\) is a measure of kinetic energy loss in the CoM-frame: \(e^{2}=T_{A 2, x}^{\prime} / T_{A 0, x}^{\prime}\)
- Collision against a wall: \(\overrightarrow{\boldsymbol{v}}_{A 2}=-e v_{A 0, x} \hat{\imath}+v_{A 0, y} \hat{\boldsymbol{\jmath}}\)