A satellite of mass m is orbiting the earth, mass M , in a circular orbit of radius ra . Unfortunately a piece of space debris left by a passing rocket lies directly in the satellite's path. The piece of debris has the same mass m as the satellite. The debris collides with the satellite and sticks to the satellite. Assume that the debris has negligible speed just before the collision. After the collision, the satellite and debris enter an elliptical orbit around the earth. The distance of closest approach to the earth of the satellite and the debris is rp . Let G be the universal constant of gravity. You may assume that M>>m .
(a) Find an expression for the speed v0 of the satellite before the collision. You may express your answer in terms of M, ra and G as needed.
v0=
(b) Calculate the ratio ra/rp .
have you got answer of any one of your questions?????
:p
(a) sqrt(G*M/r_a)
Do you have the next one too? Thanks
b) 7
To find the expression for the speed v0 of the satellite before the collision, we can use the concept of conservation of angular momentum.
(a) Conservation of angular momentum states that the total angular momentum of a system remains constant as long as no external torques act on it. In this case, the satellite is in a circular orbit before the collision, so its angular momentum is given by:
L1 = m * v0 * ra,
where v0 is the speed of the satellite, m is the mass of the satellite, and ra is the radius of the circular orbit.
After the collision, the satellite and debris enter an elliptical orbit. When the distance of closest approach to the Earth of the satellite and debris is rp, we can assume that the total angular momentum is conserved. Since the debris has negligible speed before the collision, its contribution to the angular momentum is negligible.
The moment of inertia of the satellite changed after the collision due to the addition of mass m from the debris. Considering that the debris is now stuck to the satellite, the new angular momentum is:
L2 = (m + m) * v * rp,
where v is the speed of the satellite in the elliptical orbit and rp is the distance of closest approach.
Using the conservation of angular momentum, we can equate L1 and L2:
m * v0 * ra = (2m) * v * rp.
Simplifying the equation, we get:
v0 * ra = 2v * rp.
Solving for v0, we find:
v0 = (2v * rp) / ra.
(b) To calculate the ratio ra/rp, we can rearrange the equation from part (a) and substitute v0:
v0 * ra = 2v * rp.
Substituting the expression for v0, we have:
(2v * rp) / ra * ra = 2v * rp.
Canceling out the common factors, we find:
ra = rp.
Therefore, the ratio ra/rp is equal to 1.