Conservation law Problems

Problems are from Chapter 5 of Morin

• 5.1: 120 degrees. Recall how equilibrium is related to the shortest length.
• 5.2: To calculate the overall time you can do the integral $\int dx/v$. It seems obvious but I have never had to use it.
• 5.3
• 5.4
• 5.7. Bead on a V(x) hill. A good problem that can be solved in two ways. One is much harder (I took that path) with $-g\sin (\theta )=dv/dt$, note that this is not horizontal acceleration $\dot{x}$. The second is quicker and more general.
• 5.9 Simple Oscillations
• 5.14. Snowball. Interesting qualitative result: large object pickups all momentum, but none of the energy.
• 5.15: Accelerating a car with balls. Finally derived the essential $(u-v)/u$ part.
• 5.16: Similar to 5.16, but I still made a mistake. Had to solve two differential equations separately. First, find how $m$ depends on $v$ , and then plug it into the initial $dp/dt$ equation.
• 5.17: I cooked here. Read Appendix C to understand where to use $F=ma$ and where $F=dp/dt$. In 5.16 I tried used $F=dp/dt=d(mv)/dt=vdm/dt+mdv/dt$, but it’s wrong. The reason is whenever a ball hits a car, it has a mass m at that point. Only after collision does it the mass change, so you can consider them as separate processes.