help w/ physics

Question

Astronauts in low earth orbit on the ISS are orbiting at an altitude of about 109 km above the surface of the earth. it takes about 93 minutes for an astronaut to orbit the earth once at this altitude.

A) what's the centripetal acceleration of an astronaut in the orbit described above?

B) assuming the astronaut has a mass of 75 kg, what is the magnitude of the force of earth's gravity acting on them?

C) whats the amount of kinetic energy the 75 kg astronaut has? how long would it take for a 500 W motor to provide this same amount of energy?

Answer 1

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 = v²/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 = v²/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

Answer 2

I will show you how to work this, but YOU do the calculations.

(A)

m = astronaut mass in kg

M = earth mass in kg (look it up)

G = constant of gravitation in Nm²/kg² (look it up)

h = altitude of orbit (given in the question, but convert it to meters)

R_e = radius of earth in meters (look it up)

v = √[GM / (R_e + h)]

Plug in the numbers and get v in m/s. Then

a_c = v² / (R_e + h)

Plug in the numbers and get a_c in m/s².

(B)

F = GMm / (R_e + h)²

Plug in the numbers and get F in newtons.

(C)

K = mv² / 2

Plug in the numbers and get K in joules.

P = 500 W = 500 J/s

K = Pt

t = K/P

Plug in the numbers and get t in seconds.