Notes
Momentum and impulse
- Momentum: $\vec p = m\vec v$, a vector.
- Impulse: $\vec J = \vec F_{avg}\Delta t = \Delta\vec p$. On a force vs time graph, impulse is the area.
- Spreading a collision over more time (airbags, bending your knees) reduces the average force for the same change in momentum.
- Newton's second law in momentum form: $\displaystyle \vec F_{net} = \frac{\Delta\vec p}{\Delta t}$.
Conservation of momentum
If the net external force on a system is zero, the total momentum stays constant:
- The center-of-mass velocity of an isolated system doesn't change.
- Explosions: start at rest, total momentum stays zero, so pieces move in opposite directions.
Types of collisions
| Collision | Momentum | Kinetic energy |
|---|---|---|
| Elastic | Conserved | Conserved |
| Inelastic | Conserved | Not conserved (some becomes internal energy) |
| Perfectly inelastic | Conserved | Most lost; objects stick together |
- Total energy is always conserved; only kinetic energy can change.
Practice
$p = mv = 1000(20)$.
Same impulse over a longer time means less average force.
$\Delta p = 0.15(0 - 20) = -3.0$ N·s.
$6.0 = 3.0v \Rightarrow v = 2.0$ m/s.
It's perfectly inelastic: some kinetic energy becomes internal energy.
Total momentum stays zero: $60v + 30(2.0) = 0 \Rightarrow v = -1.0$ m/s.
$J = F\Delta t$.
That's the definition of elastic.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A 0.020 kg bullet moving at 300 m/s embeds in a 2.0 kg block at rest on a frictionless table.
(a) Find the speed of the block and bullet after the collision.
Show the answer and scoring guide
$0.020(300) = 2.02v \Rightarrow v = 2.97$ m/s.
Check off the points you earned:
(b) Find the kinetic energy lost in the collision.
Show the answer and scoring guide
Before: $\displaystyle \frac{1}{2}(0.020)(300)^2 = 900$ J. After: $\displaystyle \frac{1}{2}(2.02)(2.97)^2 \approx 8.9$ J. Lost: about 891 J.
Check off the points you earned:
(c) Explain where the lost kinetic energy went.
Show the answer and scoring guide
It became internal energy (heating the bullet and block), sound, and the energy to deform the block.
Check off the points you earned: