Physics 261: Problem Set #5

1. (K+K 3.1) Just like example 3.3 in the book, only with a different density function.
2. (K+K 3.5) Split the problem into parts. The first part is to find the velocity of the acrobat JUST BEFORE he picks up the monkey. The second part is to find the velocity of the acrobat and monkey JUST AFTER the monkey has been picked up. Use conservation of momentum for this part. The third part is: Given this initial velocity and the total mass M + m, how high will the pair go?
3. (K+K 3.9) Note that the total mass of the freight car plus sand is changing with time, but you know the rate dm/dt. So you know the total mass at any time and how long it takes to run out. You can apply the rocket equation (3.19); what is the relative velocity u in this case?
   Alternative: Consider what happens at time t and then time t + delta t. The momentum change for a system consisting of the freight car and the sand in it (be sure to include the sand that is released during time delta t!) must be equal to the impulse: F*delta t. You should be able to derive a separable differential equation for dv/dt.
   In either case, you can solve the differential equation for the final velocity v, either directly or using Mathematica.
4. (K+K 3.10) To find the speed at time t, it is sufficient to find the change in momentum, if you know the final mass (what is the initial momentum?). Remember that the final mass includes the freight car mass PLUS the mass of the sand added during time t. How much does the momentum change? Calculate the impulse! (It's easy to calculate because the force is constant.)