Notes
Momentum and impulse
- Momentum $p = mv$ (vector, $\text{kg·m/s}$).
- Impulse $J = F\Delta t = \Delta p$. Bouncing back gives a bigger $\Delta p$ than stopping.
- For the same $\Delta p$, a longer time means a smaller force: air bags, helmets and bent knees.
Conservation of momentum
- In a closed system, total momentum before $=$ total momentum after: $m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$.
- It follows from Newton's third law. Watch the signs of velocities.
Types of collisions
- Elastic: momentum and kinetic energy conserved. Inelastic: momentum conserved, some kinetic energy lost. Perfectly inelastic: objects stick together.
- Explosions and recoil: total momentum stays zero, so $m_1v_1 = m_2v_2$ in opposite directions.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$|\Delta p| = 0.06 \times 50 = 3 \text{ kg·m/s} = 3 \text{ N·s}$.
The mat increases the stopping time, so the force needed for the same $\Delta p$ is smaller.
$3(4) = 4v \Rightarrow v = 3 \text{ m/s}$.
$\Delta p = 2000 \times 0.0005 = 1 \text{ kg·m/s}$; $\displaystyle v = \frac{1}{0.045} \approx 22 \text{ m/s}$.
This is a perfectly inelastic collision.
$2 \times 10 = 100v \Rightarrow v = 0.2 \text{ m/s}$, opposite the throw.
Free response
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Free-response question
On a level track, a 2,000 kg railroad car moving at $4.0 \text{ m/s}$ runs into a 3,000 kg car that is at rest. The two cars couple together and move off as one. Ignore friction.
(a) What is the total momentum of the system before the collision?
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$p = m_1v_1 + m_2v_2 = 2000(4.0) + 3000(0) = 8000 \text{ kg·m/s}$ in the direction of the moving car.
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(b) What is the velocity of the coupled cars just after the collision?
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Momentum is conserved: $8000 = (2000 + 3000)v \Rightarrow v = 1.6 \text{ m/s}$ in the original direction.
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(c) Find the kinetic energy before and after the collision. What kind of collision is this, and where did the missing energy go?
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Before: $\displaystyle \frac{1}{2}(2000)(4.0^2) = 16000 \text{ J}$. After: $\displaystyle \frac{1}{2}(5000)(1.6^2) = 6400 \text{ J}$. $9600 \text{ J}$ of kinetic energy was lost, so the collision is perfectly inelastic. The energy became sound, heat and bending of the coupling.
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