Final answer:
To find the speed of the satellite at apogee, we can use the principle of conservation of angular momentum. By equating the angular momentum of the satellite before and after the collision, and using the equations for circular motion and the distance at apogee, we can calculate that the speed of the satellite at apogee is 3.21 km/s.
Step-by-step explanation:
To find the speed of the satellite at apogee, we can use the principle of conservation of angular momentum. Angular momentum is given by the equation: L = mvr, where L is the angular momentum, m is the mass of the object, v is its velocity, and r is the distance from the center of rotation. Since the angular momentum is conserved, we can equate the angular momentum of the satellite before and after the collision:
msvsrs = mavara
Where ms is the mass of the satellite, vs is its velocity before the collision, rs is its distance from the center of the Earth before the collision, ma is the mass of the asteroid, va is the velocity of the satellite after the collision, and ra is the distance from the center of Earth at apogee.
Since the satellite is in a circular orbit before the collision, its velocity before the collision is given by the equation: vs = √ (GM/rs), where G is the gravitational constant and M is the mass of the Earth.
At apogee, the distance from the center of Earth is given as 45,000.0 km, so we can substitute the known values into the equation to find the velocity of the satellite at apogee:
va = √ (GM/ra)
Using the given values for ra and solving the equation, we find that the speed of the satellite at apogee is 3.21 km/s.