Space debris left from old satellites and their launchers is becoming a hazard t

Question

Space debris left from old satellites and their launchers is becoming a hazard to other satellites.

(a) Calculate the speed of a satellite in an orbit 800 km above the earth's surface.
? m/s
(b) Suppose a loose rivet is in an orbit of the same radius that intersects the satellite's orbit at an angle of 90°. What is the velocity of the rivet relative to the satellite just before striking it?
? m/s
(c) Given the rivet is 5.00 mm in size, how long will its collision with the satellite last? (Assume that the rivet flattens out like a pancake when it splats onto the satellite.)
? s
(d) If its mass is 0.100 g, what is the average force it exerts on the satellite?
? N
(e) How much energy in joules is generated by the collision? (The satellite's velocity does not change appreciably, because its mass is much greater than the rivet's.)
? J

Explanation / Answer

a)

here

Fc = m * ac

a = v^2 / r and F = G * m * M / r^2

GmM /r^2 = mv^2 / r

v = sqrt( G*M / r)

v = sqrt( (6.673 * 10^-11 * 5.979 * 10^24) / (800 * 10^3))

v = 2.23 * 10^4 m/s

b)

vtot = sqrt( v^2 + v^2) = sqrt(2*v^2)

vtot = sqrt(2) * 2.23 * 10^4 = 3.15 * 10^4 m/s

c)

using kinematics equation

d = vtot * t

t = d / vtot

t = 5 * 10^-3 / 3.15 * 10^4

t = 1.58 * 10^-7 sec

d)

F = delta p / delta t = m * vtot / t

F = (0.1 * 10^-3 * 3.15 * 10^4) / (1.58* 10^-7)

F = 0.199 * 10^8 N

e)

the energy is generated form the rivet in the satellite's frame of reference , vi = vtot and vf=0

so the change in the kinetic energy of the rivet is

delta KE = 0.5 * m * vtot^2 - 0.5 * m * vi^2

delta KE = 0.5 * 0.1 * 10^-3 * (3.15 * 10^4) ^2 - 0

delta KE = 4.96 *10^4 J

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