Heidi T. answered • 12/01/19

MA in Mathematics, PhD in Physics with 7+ years teaching experience

Before solving this problem, you need to collect some information:\

Earth's radius = 6371 km

Earth's mass = 5.972 X 10^24 kg

Gravitational Constant: 6.67408 x 10^(-11) m^3 kg^(-1) s^(-2)

A) The equation for the centripetal acceleration is a_{c} = v^{2}/r , where v is the tangential velocity (speed in orbit) and r is the distance from the center of the orbit (center of Earth)

r = R_{earth} + h = 6371 km + 109 km = 6480. km = 6.480 x 10^6 m

Speed is distance/time, so the speed is the distance traveled in one orbit,

C = 2πr = 2π(6.480 x 10^6 m) = 4.0715 X 10^7 m

The time is given as 93 minutes = 5580 s

==> v = (4.0715 X 10^7 m) / (5580 s) = 7.297 X 10^3 m/s

You might have been able to give the answer in km/min, which is less calculation now, but would need to convert to m/s before the final answer.

Now that v and r have been calculated, the centripetal acceleration can be calculated.

a_{c} = v^{2}/r = (7.297 X 10^3 m/s)^{2}/( 6.480 x 10^6 m) = 8.216 m/s^2

B) Because the astronaut is above the surface of the Earth, the gravitational force will be less than on the surface of the earth, so g = 9.81 m/s^2 can't be used. Use the universal gravitational force equation:

F = G M m/r^2 = (6.67408 x 10^(-11))(5,972 x 10^24 kg)(75 kg) / (6.480 x 10^6 m)^2 = 711.9 N

C) The kinetic energy of the astronaut is

K = (1/2) m v^2 = (1/2) (75 kg) (7.297 X 10^3 m/s)^{2} = 1.996 x 10^9 N

Power, P = W/t where W == work = ΔK, so P = ΔK / t

rearranging to solve for time,

t = ΔK / P = (1.996 x 10^9 N) / 500 W = 3.993 x 10^6 s