Escape and Binding Energies
Ep = mgh
Fg = mg
- this only applies where h << rE
In general, Ep is due to the force of gravity on on object:
Fg = G mo me
r2
Graphically:
Fg
(N)
Area has the units of Nm = J
Centre of Earth
re



r (m)
Shaded Area: W = Ep
The shaded area is the work that must be done in moving an object from
the earth’s surface to infinity.
We can calculate this area using calculus:
Shaded Area =


Fg (r) dr



G m o me

r2
=
re


dr
re
=-



G mo me

r
r

e
= - 0 - G mo me
re
= G mo me
re
Therefore the Work to move an object from the surface of the earth to
infinity is
W= G mo me
re
If we define the potential energy of an object at distance to be zero,
then the Gravitation Ep at the surface of the earth is:
EP at Earths Surface

Note : E total, Ep and Ek
=
- G mo me
re
are all zero at infinity.
In general: The potential energy of any object at any distance from the
earth will always be negative.
EP
=
- G mo me
r
Measure from the
centre of the earth
Question 1
How much Ek is required to launch a rocket (mo) from the Earth’s surface
and send it to infinity?
The Ep of the rocket must change from
to escape.
- Gmome to 0 for the rocket
re
Therefore the following energy must be given to the rocket to escape:
Ek escape = G mo me
re
(we already knew this from the bottom of page 1.)
Question 2
What velocity is required so that the rocket will escape?
Ek escape = G mo me
re
½ m0 v 2 = G m o me
re
vescape =
2Gme
re
vescape = 1.12 x 104 m/s
Question 3
What is the total energy of an orbiting object?
Let ro be the radius of the orbit.
Etotol (orbit) = Ek(orbit) + Ep (orbit)
Etotol (orbit) = ½ movo2 - Gmemo
ro
Also,
[1]
Fc = F g
movo2 = Gmemo
ro
ro2
movo2 = Gmemo
ro
½ movo2 = ½ Gmemo
ro
Multiple by ½
Subsitute [2] into [1]:
E total (orbit) = 1 Gmemo
2 ro
- Gmemo
ro
E total (orbit) = - 1 Gmemo
2 ro
[2]
Question 4
What additional energy must be given to a rocket to escape an orbit?
The total energy of an object in orbit is given by:
E total (orbit) = - 1 Gmemo
2 ro
Therefore to send the rocket to the amount of energy required is:
E k (binding) =
1 Gmemo
2 ro
(this would bring the total energy to zero at infinity)
Question 5 (the Last one)
What energy must be given to a rocket to put it in orbit?
Etotal (orbit) = Ek (initial on earth) + Ep (initial on Earth)
-1 Gmemo = Ek (initial on earth) - Gmemo
2 ro
re
Ek (initial) = Gmemo - 1 Gmemo
re
2 ro
Summary
The potential energy of an object at any distance from the Earth is given
by:
EP
=
- G mo me
r
The total energy of an object the Earth is given by:
E total (orbit) = - 1 Gmemo
2 ro
The binding energy of any mass is the amount of additional energy it
requires for escape to infinity.
Binding Energy from Earth:
Ek = Gmemo
re
Binding Energy from orbit:
Ek=
1 Gmemo
2 ro
The Energy required to put an object into orbit:
Ek (initial) = Gmemo re
1 Gmemo
2 ro
Escape / Binding Energy Questions
mo
EARTH
Re
in orbit
a)
5Re
c)
3Re
b)
v=0
not in
orbit
Determine an equation for each of the following:
1) The potential energy of each position a),b) and c).
2) The energy required to move mo from a) to b).
3) The velocity required to move mo from a) to b)
4) The energy required to move mo from a) to c).
5) The velocity required to move mo from a) to c).
6) The energy required to escape from a), b) and c)