- #1

Yalanhar

- 39

- 2

- Homework Statement
- Two planets A and B (mass m for both) are separated by a distance of D. A object is thrown from the surface of planet A towards planet B. Radius of planet A and B are R and 2R. Find the minimum velocity to throw the object from planet A to B.

- Relevant Equations
- ##dW=\frac {GMm}{r^2}dr##

##W=\Delta U_p##

##K_o +U_o = K+U##

So I integrated the work done on the object by both planets. Work1 is until x, and Work2 is from x to d. Where x is the point where both gravitational forces are equal.

##W_1=\int_0^x \frac{GMm}{r^2}dr - \int_0^x \frac{GMm}{(3R+D-r)^2}dr ##

##W_2=\int_x^D \frac{GMm}{(3R+D-r)^2}dr - \int_x^D \frac{GMm}{r^2}dr ##

##W_t=W_1+W_2##

##W_t = \Delta U_p##

Chosing zero at A's surface

##W_t=U##

And then I have to conserve the energy and calculate ##v_o##. However, the algebraic is a pain and I still have that ##x=\frac{3R+D}{2}##.

Any help? The answer is ##v_o = \sqrt \frac{GM(D-R)}{R(3R+D)} ##

##W_1=\int_0^x \frac{GMm}{r^2}dr - \int_0^x \frac{GMm}{(3R+D-r)^2}dr ##

##W_2=\int_x^D \frac{GMm}{(3R+D-r)^2}dr - \int_x^D \frac{GMm}{r^2}dr ##

##W_t=W_1+W_2##

##W_t = \Delta U_p##

Chosing zero at A's surface

##W_t=U##

And then I have to conserve the energy and calculate ##v_o##. However, the algebraic is a pain and I still have that ##x=\frac{3R+D}{2}##.

Any help? The answer is ##v_o = \sqrt \frac{GM(D-R)}{R(3R+D)} ##