A 1300-kg space vehicle falls vertically from a height of 2500 km above the earth's
surface. Determine how much work is done by the force of gravity in bringing the vehicle
to the earth's surface.
I assume you have calculus. g is changing, so we have to use calculus.
Work= INT Force*dx
= INT GM3m/distance^2 dx integrated from radEarth to 2500km+radEarth.
For such a large change in altitude, the change of the acceleratin of gravity as it falls must be taken into account.
The Earth's radius is 6370 km. The change in potential energy falling from altitude 2500 km to the surface is
G M m (1/6370*10^3 - 1/8870*10^3 m)
where m = 1300 kg, M is the mass of the Earth and G is the universal constant of gravity. If you use the fact that GM/Re^2 = g, where Re is the Earth's radius and g is the acceleration of gravity there, then the P.E. change can be written
delta PE = m g Re^2[1/Re - 1/8870*10^3]
= m g Re[1 - Re/(Re+2500*10^3)]
= 1300*9.8*6370*10^3[1 - (6370/8870)]
= 2.29*10^10 Joules
To determine the work done by the force of gravity in bringing the vehicle to the Earth's surface, we can use the formula:
Work = Force * Distance
The force acting on the vehicle is the gravitational force, which can be calculated using Newton's law of universal gravitation:
F = G * (m1 * m2) / r^2
Where:
G is the gravitational constant (approximately 6.67430 × 10^-11 N*m^2/kg^2)
m1 is the mass of the vehicle (1300 kg)
m2 is the mass of the Earth (5.972 × 10^24 kg, which is the approximate mass of the Earth)
r is the distance between the center of the Earth and the vehicle (the sum of the radius of the Earth and the initial height of the vehicle)
The distance the vehicle falls is the initial height of the vehicle, which is 2500 km (or 2,500,000 meters).
Now let's calculate step by step:
Step 1: Calculate the distance between the center of the Earth and the vehicle.
The radius of the Earth is approximately 6,371 km (or 6,371,000 meters).
So, the total distance is:
r = radius of the Earth + height of the vehicle
r = 6,371,000 + 2,500,000
r = 8,871,000 meters
Step 2: Calculate the gravitational force.
F = (6.67430 × 10^-11) * (1300 * 5.972 × 10^24) / (8,871,000)^2
F = 9.75458 N (approximately)
Step 3: Calculate the work done.
Work = Force * Distance
Work = 9.75458 * 2,500,000
Work ≈ 24,386,450 Joules
Therefore, the work done by the force of gravity in bringing the vehicle to the Earth's surface is approximately 24,386,450 Joules.
To determine how much work is done by the force of gravity in bringing the vehicle to the earth's surface, we need to find the gravitational potential energy difference between the initial height and the surface of the Earth.
The formula for gravitational potential energy is given by:
Potential energy (PE) = mgh
Where:
m = mass of the space vehicle = 1300 kg
g = acceleration due to gravity on Earth = 9.8 m/s^2
h = height difference = 2500 km = 2500000 m
First, we need to calculate the initial potential energy (PE_initial) at the height of 2500 km above the Earth's surface:
PE_initial = m * g * h_initial
= 1300 kg * 9.8 m/s^2 * 2500000 m
= 3.215 × 10^10 Joules
Next, we calculate the final potential energy (PE_final) at the surface of the Earth:
PE_final = m * g * h_final
= 1300 kg * 9.8 m/s^2 * 0 m
= 0 Joules
The work done by the force of gravity can be determined by taking the difference between the initial and final potential energy:
Work done = PE_final - PE_initial
= 0 - 3.215 × 10^10 Joules
= -3.215 × 10^10 Joules
Therefore, the work done by the force of gravity in bringing the vehicle to the Earth's surface is approximately -3.215 × 10^10 Joules. The negative sign indicates that work is done by the gravitational force as the vehicle falls downward.