1. Inelastic Collisions
As you come in, please set clicker to channel 44 and then answer the following question (before the lecture starts).
Quiz – suppose rain falls vertically into an open cart rolling along a straight horizontal track with negligible friction. As a result of the accumulating water, the speed of the cart
increases
does not change
decreases

2. Topics for Today A bit more on inelastic collisions (9-6)
Elastic collisions in one dimension (9-7)
Collisions in two dimensions (9-8)
Systems with varying mass: a rocket (9-9)

3. Inelastic Collisions Do inelastic collision on air track.
Quiz – a Prius (mass = 1300 kg) and a Chevy Suburban (mass = kg) are traveling towards each other at 120 km/hr. They collide and stick together with a speed of 40 km/hr. About 2×106 J of energy is lost in the collision. Where does it go?
Into heat (thermal energy)
Into internal energy
Into sound waves
All of the above ↓

4. Elastic Collisions
In elastic collisions, kinetic energy is conserved as well as momentum.
Let’s do the equations in one dimension.
Momentum:
𝑃 𝑖 = 𝑝 1𝑖 + 𝑝 2𝑖 = 𝑚 1 𝑣 1𝑖 + 𝑚 2 𝑣 2𝑖 = 𝑃 𝑓 = 𝑝 1𝑓 + 𝑝 2𝑓 = 𝑚 1 𝑣 1𝑓 + 𝑚 2 𝑣 2𝑓
𝑚 1 𝑣 1𝑖 + 𝑚 2 𝑣 2𝑖 = 𝑚 1 𝑣 1𝑓 + 𝑚 2 𝑣 2𝑓
Kinetic energy:
𝐾 𝑖 = 𝑚 1 𝑣 1𝑖 𝑚 2 𝑣 2𝑖 2 = 𝐾 𝑓 = 𝑚 1 𝑣 1𝑓 𝑚 2 𝑣 2𝑓 2
We assume that the initial state is known: m1, m2, v1i, v2i.
The final state has two unknown variables, v1f and v2f.
Since there are two equations and two unknowns, we can solve.

5. Elastic Collisions Rewrite 𝑚 1 𝑣 1𝑖 + 𝑚 2 𝑣 2𝑖 = 𝑚 1 𝑣 1𝑓 + 𝑚 2 𝑣 2𝑓
𝑚 1 𝑣 1𝑖 − 𝑣 1𝑓 = − 𝑚 2 (𝑣 2𝑖 − 𝑣 2𝑓 )
Rewrite 𝑚 1 𝑣 1𝑖 𝑚 2 𝑣 2𝑖 2 = 𝑚 1 𝑣 1𝑓 𝑚 2 𝑣 2𝑓 2
𝑚 1 𝑣 1𝑖 − 𝑣 1𝑓 𝑣 1𝑖 + 𝑣 1𝑓 =− 𝑚 2 (𝑣 2𝑖 − 𝑣 2𝑓 ) (𝑣 2𝑖 +𝑣 2𝑓 )
Divide the second equation by the first one, do algebra. Find:
𝑣 1𝑓 = 𝑚 1 − 𝑚 2 𝑚 1 + 𝑚 2 𝑣 1𝑖 + 2𝑚 2 𝑚 1 + 𝑚 2 𝑣 2𝑖
𝑣 2𝑓 = 2 𝑚 1 𝑚 1 + 𝑚 2 𝑣 1𝑖 + 𝑚 2 − 𝑚 1 𝑚 1 + 𝑚 2 𝑣 2𝑖

6. Elastic Collisions
Quiz – Object 1 starts with velocity +v, object 2 starts at rest, masses are equal. What are the velocities of the two masses after the collision?
𝑣 1 =+𝑣, 𝑣 2 =0
𝑣 1 =0, 𝑣 2 =+𝑣
𝑣 1 =+ 𝑣 2 , 𝑣 2 =+𝑣/2
𝑣 1 =−𝑣, 𝑣 2 =0
Do this collision on air track.

7. Elastic Collisions
Quiz – Object 1 starts with velocity +v, object 2 starts at rest, object 2 is much more massive than object 1. What are the velocities of the two masses after the collision?
𝑣 1 =+𝑣, 𝑣 2 =0
𝑣 1 =0, 𝑣 2 =+𝑣
𝑣 1 =+ 𝑣 2 , 𝑣 2 =+𝑣/2
𝑣 1 =−𝑣, 𝑣 2 =0
Do this collision on air track.

8. Elastic Collisions
Quiz – Object 1 starts with velocity +v, object 2 starts at rest, object 1 is much more massive than object 2. What are the velocities of the two masses after the collision?
𝑣 1 =+𝑣, 𝑣 2 =0
𝑣 1 =0, 𝑣 2 =+𝑣
𝑣 1 =𝑣, 𝑣 2 =+2𝑣
𝑣 1 =−𝑣, 𝑣 2 =0
Do this collision on air track.

9. Elastic Collisions Now let’s do the equations in two dimensions.
Kinetic energy is the same (v’s are magnitudes of velocity):
𝐾 𝑖 = 𝑚 1 𝑣 1𝑖 𝑚 2 𝑣 2𝑖 2 = 𝐾 𝑓 = 𝑚 1 𝑣 1𝑓 𝑚 2 𝑣 2𝑓 2
There are now two momentum equations, one for each axis:
𝑚 1 𝑣 1𝑖𝑥 + 𝑚 2 𝑣 2𝑖𝑥 = 𝑚 1 𝑣 1𝑓𝑥 + 𝑚 2 𝑣 2𝑓𝑥
𝑚 1 𝑣 1𝑖𝑦 + 𝑚 2 𝑣 2𝑖𝑦 = 𝑚 1 𝑣 1𝑓𝑦 + 𝑚 2 𝑣 2𝑓𝑦
Initial state is known: m1, m2, v1ix, v1iy. v2ix, v2iy
The final state has four unknown variables, v1fx, v1fy. v2fx, v2fy
There are three equations and four unknowns, therefore there is no unique solutions. We need one more parameters to set the final state.

10. Elastic Collisions
In pool, this allows us to set the angle between the outgoing balls.

11. Rockets
In a rocket, the mass of the rocket changes as it ejects propellant. Momentum is still conserved.
Initial momentum 𝑃 𝑖 =𝑀𝑣
Rocket ejects propellant with mass –dM.
Mass of rocket changes to M+dM, speed to v+dv
Velocity of propellant 𝑈=𝑣+𝑑𝑣− 𝑣 𝑟𝑒𝑙 where vrel is the relative velocity of propellant vs rocket.
𝑃 𝑓 =−𝑑𝑀 𝑣+𝑑𝑣− 𝑣 𝑟𝑒𝑙 + 𝑀+𝑑𝑀 𝑣+𝑑𝑣 = 𝑃 𝑖 =𝑀𝑣
−𝑑𝑀𝑣−𝑑𝑀𝑑𝑣+𝑑𝑀 𝑣 𝑟𝑒𝑙 +𝑀𝑣+𝑀𝑑𝑣+𝑑𝑀𝑣+𝑑𝑀𝑑𝑣= 𝑀𝑣
𝑀𝑑𝑣=− 𝑣 𝑟𝑒𝑙 𝑑𝑀
Divide by dt: 𝑀 𝑑𝑣 𝑑𝑡 =𝑀𝑎=− 𝑣 𝑟𝑒𝑙 𝑑𝑀 𝑑𝑡

12. Rockets Do demo 1N22.10, Rocket Car - Fire Extinguisher.
Acceleration: 𝑀𝑎=− 𝑣 𝑟𝑒𝑙 𝑑𝑀 𝑑𝑡

13. Rockets
Since the mass of the rocket changes with time, we won’t have the usual v = at relation. We need to integrate.
𝑀𝑑𝑣=− 𝑣 𝑟𝑒𝑙 𝑑𝑀
𝑑𝑣=− 𝑣 𝑟𝑒𝑙 𝑑𝑀 𝑀
𝑣 𝑖 𝑣 𝑓 𝑑𝑣 =− 𝑀 𝑖 𝑀 𝑓 𝑣 𝑟𝑒𝑙 𝑀 𝑑𝑀
𝑣 𝑓 − 𝑣 𝑖 = −𝑣 𝑟𝑒𝑙 ln 𝑀 𝑓 𝑀 𝑖 = 𝑣 𝑟𝑒𝑙 ln 𝑀 𝑖 𝑀 𝑓
To make an efficient rocket, one wants high vrel. Otherwise, one needs to eject a lot of mass (high Mi versus Mf).

14. Rockets
Quiz – The Saturn V rocket that took men to the Moon was built in stages that were dropped off during flight. Why?
For convenience
To reduce the litter left on the Moon.
To maximize vrel
To maximize Mi versus Mf

15. Rockets
Quiz – The Saturn V rocket that took men to the Moon was built in stages that were dropped off during flight. Why?
For convenience
To reduce the litter left on the Moon.
To maximize vrel
To maximize Mi versus Mf

16. Rockets
Quiz – The Saturn V rocket that took men to the Moon was built in stages that were dropped off during flight. Why?
For convenience
To reduce the litter left on the Moon.
To maximize vrel
To maximize Mi versus Mf