Variables
We will derive the escape speed by comparing the potential and kinetic energies of the
rocket as it blasts off (subscript 0), and as it approaches infinity (subscript ).
Strategy
- Calculate the change in potential energy of the Earth-rocket system between
launch and an infinite separation of the two. - Calculate the change in kinetic energy between launch and an infinite separation.
- The conservation of mechanical energy states that the total energy after the
engines have ceased firing equals the total energy at infinity. State this
relationship and solve for v, the initial escape speed.
Physics principles and equations
We use the equation for the potential energy of an object at a distance r from the center
of the planet.
We use the definition of kinetic energy.
Finally, we will use the equation that expresses the conservation of mechanical energy.
ǻPE + ǻKE = 0
Step-by-step derivation
In the first steps we find the potential energy of the rocket at launch and at infinity, and
subtract the two values.
In the following steps we find the kinetic energy of the rocket at launch and at infinity, and subtract the two values.
Escape speed equation
v = escape speed
G = gravitational constant
M = mass of planet
r = radius of planet
What is the minimum speed
required to escape the Earth’s
gravity?
v = 11,200 m/s
initial final change
potential energy PE 0 PE ǻPEkinetic energy KE 0 KE ǻKE
gravitational constant G
mass of planet M
mass of rocket m
radius of planet r
initial speed of rocket v