Question

Space debris left from old satellites and their launchers is becoming a hazard to other satellites. (a)Calculate the speed in m/s of a satellite in an orbit 860 km above the Earth's surface. (b) Suppose a loose rivet is in an orbit of the same radius that intersects the satellite's orbit at an angle of 90° relative to Earth. What is the velocity in m/s of the rivet relative to the satellite just before striking it? (Enter the magnitude.) (c) Given the rivet is 2.00 mm in size, how long in seconds will its collision with the satellite last? (Assume the rivet is flattened like a pancake on the surface of the satellite during the collision.) (d) If its mass is 0.350 g, what is the average force in newtons it exerts on the satellite? (Enter the magnitude.) (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.)

Answer 1

(a)

M_e = mass of earth = 5.98 x 10²⁴ kg

R = radius of earth = 6.4 x 10⁶ m

h = height above earth's surface = 860 km = 0.86 x 10⁶ m

r = radius of the orbit = R + h = (6.4 x 10⁶) + (0.86 x 10⁶) = 7.26 x 10⁶ m

v = orbital speed of satellite

orbital speed is given as

v = sqrt(G M_e/r)

v = sqrt((6.67 x 10^-11) (5.98 x 10²⁴)/(7.26 x 10⁶))

v = 7412.2 m/s

b)

v_rivet = sqrt(2) v

v_rivet = sqrt(2) (7412.2)

v_rivet = 10482.4 m/s

c)

d = size = 2 mm = 0.002 m

t = time for collision = ?

time for collision is given as

t = d/v_rivet

t = 0.002/10482.4 = 1.9 x 10^-7 sec

d)

using impulse change in momentum equation

F t = m v_rivet

F (1.9 x 10^-7) = (0.35 x 10^-3) (10482.4)

F = 1.93 x 10⁷ N

e)

energy generated = (0.5) m v²_rivet  = (0.5) (0.35 x 10^-3) (10482.4)² = 1.92 x 10⁴ J