Download The total mechanical energy U of the rocket is

The total mechanical energy U of the rocket is composed of two components:
1. The kinetic energy E due to the velocity of the rocket.
2. The potential energy V of the rocket with respect to earth’s center.
The principle of conservation of energy states that the total mechanical energy in the
closed system remains constant:
U  E  V  const.
The kinetic energy of the rocket as a function of its speed v and mass M is:
E
Mv 2
2
While the potential energy as a function of its mass and distance r from the earth’s center
is given by:
GMM e
V 
r
Where G is the gravitational constant and Me is the mass of earth.
At point a (where the engines shut off) the total mechanical energy is:
U a  E a  Va 
Mv a2 GMM e

2
Re  d a
Where Re is earth’s radius and da is the height of the rocket above earth’s surface.
At point b which is 1000 km above earth’s surface, the total mechanical energy is:
U b  Eb  Vb  Eb 
GMM e
Re  d b
Since the total energy must be conserved at all times we have:
Ub  Ua
Eb 
GMM e Mv a2 GMM e


Re  d b
2
Re  d a
Mv a2 GMM e GMM e
Eb 


2
R e  d a Re  d b
 1
Mv a2
1 

Eb 
 GMM e 

2
R

d
R

d
a
e
b 
 e
All that is left is to substitute the numbers:
M  150kg
v a  3.7 km s  3700 m s
G  6.67  10 11 Nm 2 kg 2
Re  6400km  6.4  10 6 m
M e  6.0  10 24 kg
d a  200km  2  10 5 m
d b  1000km  10 6 m
Eb 
 1
Mv a2
1 

 GMM e 

2
R

d
R

d
a
e
b 
 e
Eb 
150  3700 2
1
1


 6.67  10 11  150  6.0  10 24  

6
5
6
6 
2
6.4  10  10 
 6.4  10  2  10




Eb  4.3  10 7 J
The maximal altitude is obtained when the rocket is (temporarily) motionless. This means
that at this point (c), the only component of the mechanical energy is the potential energy.
Thus we get:
Uc  Ub

GMM e
GMM e
 Eb 
Re  d c
Re  d b
GMM e GMM e

 Eb
Re  d c R e  d b
GMM e GMM e  Eb Re  d b 

Re  d c
Re  d b
Re  d c
Re  d b

GMM e GMM e  Eb Re  d b 
dc 
GMM e Re  d b 
 Re
GMM e  Eb Re  d b 
And again, all that is left is to plug in the numbers:
M  150kg
G  6.67  10 11 Nm 2 kg 2
Re  6400km  6.4  10 6 m
M e  6.0  10 24 kg
d b  1000km  10 6 m
Eb  4.3  10 7 J
GMM e  0.6  1017
Re  d b  7.4  10 6
dc 
GMM e Re  d b 
 Re
GMM e  Eb Re  d b 
dc 
0.6  10  7.4  10 
0.6  10   4.3  10  7.4  10   6.4  10
17
17
6
7
6
The maximal height will be 1040km above earth’s surface.
6
 1.04  10 6 m