- #1

zachary570

- 5

- 4

- Homework Statement
- An experiment is performed in deep space with two uniform spheres, one with mass ##m_{A}## and the other with mass ##m_{B}##. They have equal radii, ##r##. The spheres are released from rest with their centers a distance ##r_{i}## apart. They accelerate toward each other and. you may ignore all gravitational forces other than that between the two spheres. When their centers are a distance ##r_{f}## apart what is the speed of each sphere?

- Relevant Equations
- Conservation of Energy: ##K_{i} + U_{i} = K_{f} + U_{f}##

Conservation of Momentum: ##m_{A}v_{A} = m_{B}v_{B}##

Gravitational Potential Energy: ##U = \frac{-Gm_{A}m_{B}}{r}##

I was working on this problem but after getting to the answer I questioned the methods that I used for previous problems that I had solved. I understand that the total energy of the system remains constant and that we use the conservation of momentum to relate the two velocities. This gives two equations with two unknowns and the math is straight forward. However, when I first attempted this I didn't realize that you need to include the kinetic energy of both objects and went about solving it the usual way where that mass cancels and I got $$v_{A} = \sqrt{2Gm_{B}\left( \frac{1}{r_{f}} - \frac{1}{r_{i}} \right)} $$ I understand now that the gravitation potential energy refers to the system of the two objects as it has the mass of both objects in the equation. What I am confused about is when it is appropriate to leave out the kinetic energy of one of the objects. Is it just when one of the objects has a much larger mass than the other object so that the acceleration the larger mass has is so small that it is negligible? What would be best when working through a problem like this on a test and I need to show my work. Should I include the larger mass's kinetic energy when expanding ##K_{i} + U_{i} = K_{f} + U_{f}## but then indicate that its velocity is approximately zero and cross it out? I only say this because on a previous problem where I was asked to find the speed of a small object when it hits the Earth after being released from rest at a large height above the Earth's surface I used the same method and got the right answer without realizing that I didn't really set the problem up correctly. Any advice would be appreciated.

Thanks,

zachary570

Thanks,

zachary570