Review
On Monday, we talked about the energy (work) required to launch an 8,700 kg object to an
altitude of 1.80 Earth radii above the surface.
GPEsurface = -5.44 × 1011 J
GPE1.80R = -1.94 × 1011 J
W = change in PE = GPE1.80R – GPEsurface = 3.5 × 1011 J
If we were launching it as a projectile (as opposed to a rocket with fuel), we could find the
launch velocity using conservation of energy: kinetic energy at takeoff equals potential energy
gained during the ascent:
KEsurface = change in PE
v = 8,970 m/s
Example Problems
Let’s use gravitational potential to find the change in energy of some different spacecraft as they
take off from an unknown planet.
1. The graph below shows the variation of the gravitational potential due to a planet with
distance r. Using the graph, estimate: (a) the gravitational potential energy of an 800 kg
spacecraft that is at rest on the surface of the planet; (b) the work done to move this
spacecraft from the surface of the planet to a distance of four planet radii from the surface
of the planet.
V /106 J kg-1
0
0
-1
-2
-3
-4
-5
r /R
1
2
3
4
5
6
7
8
2. The figure below shows the variation of the gravitational potential due to a planet and its
moon with distance r from the center of the planet. The center-to-center distance
between the planet and the moon is d. The planet’s center is at r = 0 and the center of the
moon is at r = d. What is the minimum energy required so that a 500 kg probe at rest on
the planet’s surface will arrive on the moon.
V/106 J kg-1
0
0.2
0.4
0.6
0.8
1
r/d
-50
-100
-150
-200
-250
-300
-350
-400
The benefit of the gravitational potential graphs is that they don’t depend on the mass of the
spacecraft – the graph above is a property of the planet and the moon, the two objects creating
the gravitational field the spacecraft travels through. We can use it to calculate the energy
required to launch any mass of spacecraft to the moon.
Depends on Planet(s) and
Mass of Spacecraft
Depends Only on Planet(s)
Creating the Field
Force
Field
Potential Energy
Potential
The stuff on the right is a property of the underlying setup, and when we decide to launch a
specific spacecraft we can use its mass m to calculate the stuff on the left (such as energy
required to launch it into space).
3. What is the maximum altitude attained by a 100-kg projectile shot up from the surface of
the Earth at a speed of 1.5 km/s (ignoring the atmosphere)?
4. What is the maximum altitude attained by a 100-kg projectile shot up from the planet in
problem 1?
Escape Velocity
5. What speed would you have to shoot a projectile up to completely overcome the earth’s
gravitational field (i.e. to reach 0 G.P.E. – at “very great” or infinite separation)?
This quantity is called the escape velocity of the planet and, remarkably, does not depend on the
mass of the projectile being launched. (Try it with a different mass for the projectile.)
6. Calculate the escape velocity from the surface of the sun. (Solar mass = 1.9891 x 10 31 kg,
solar radius = 6.955 x 108 m.)
7. Can you come up with a general formula for the escape velocity at the surface of a planet
of mass M and radius R?
Black Holes
Interesting note is that the radius of a black hole (or the distance from the center of the black
hole to the event horizon) can be found by using the speed of light as the escape velocity.
8. How small (i.e. what radius) would you have to make a star with the mass of the sun to
make the escape velocity at the surface of the star equal to the speed of light?
This is called the Schwarzchild radius and is the size a star of a given mass would have to
collapse down to in order to become a black hole.
Orbital Energy
So far we have been finding only the energy required to launch a particle straight up, or the
potential energy of a particle a given height in space. Now we’re going to calculate the kinetic
energy as well as the potential energy of an orbiting satellite.
Example: A global positioning satellite with a mass of 1,630 kg orbits the earth in uniform
circular motion at an altitude of 20,350 km above the earth’s surface.
(a) What is the satellite’s potential energy?
(b) At what speed is it orbiting the planet (using centripetal force equation)?
(c) Therefore, what is its kinetic energy?
Given its potential and kinetic energy, what is its total energy? Do you notice anything surprising
about that figure?
Example (cont): Finally, what is the energy required to launch the satellite into orbit (not just
straight up to that height, but into orbit with the total energy we just calculated)?