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 .
ra/rp=
rp = ra/7
(a) To find the expression for the speed v0 of the satellite before the collision, we can use the conservation of angular momentum. Angular momentum is conserved when there is no external torque acting on a system. In this case, the only external force acting on the satellite-debris system is gravity, which does not produce a torque.
The angular momentum of an object in circular motion can be expressed as the product of its moment of inertia and angular velocity. For a satellite in a circular orbit, the moment of inertia is given by I = m * ra^2, where m is the mass of the satellite and ra is the radius of the orbit.
The angular velocity of the satellite in a circular orbit can be calculated as the speed of the satellite divided by the radius of the orbit, v0 / ra.
Therefore, the angular momentum before the collision is given by L = I * (v0 / ra) = m * ra^2 * (v0 / ra) = m * ra * v0.
After the collision, the satellite and debris form an elliptical orbit. The angular momentum is conserved, so we can equate the initial angular momentum to the final angular momentum.
Before the collision: L = m * ra * v0
After the collision: L = (m + m) * rp * vf,
where rp is the distance of closest approach to the Earth and vf is the final velocity of the satellite and debris.
Setting these two expressions for angular momentum equal, we get:
m * ra * v0 = 2m * rp * vf.
Canceling the mass m and rearranging the equation, we have:
v0 = 2 * rp * vf / ra.
(b) To calculate the ratio ra / rp, we need to determine the relationship between the velocities of the satellite before and after the collision.
From the equation above, we can rearrange it to solve for vf:
vf = (v0 * ra) / (2 * rp).
The speed v0 before the collision is the same as the speed of the satellite in the circular orbit. This speed can be calculated using the formula for the orbital speed of a satellite in a circular orbit:
v0 = sqrt(G * M / ra).
Substituting this expression for v0 into the equation for vf, we have:
vf = (sqrt(G * M / ra) * ra) / (2 * rp).
Now we can calculate the ratio ra / rp:
ra / rp = ra / (rp * vf / v0).
Substituting the expressions for v0 and vf, we get:
ra / rp = ra / (rp * ((sqrt(G * M / ra) * ra) / (2 * rp))).
Simplifying the expression, we find:
ra / rp = 2 * sqrt(G * M * rp) / ra.