F=−GMmr2
Where M and m are the masses of the two objects, r is the distance between them, and G is the gravitational constant.
You can express it as a vector (in fact if you are simulating motion in a gravitational field you need to express it as a vector):
F=−GMmr2ˆr
where ˆr is the unit vector in the direction from the mass M to m.
Gravitational Potential Energy
Gravitational potential energy is defined as the work done to bring an object from a distance r to infinity.
U=∫∞rFdr
U=∫∞r−GMmr2dr
U=−GMm∫∞rr−2dr
U=−GMm[−r−1]∞r
U=GMm[1r]∞r
U=GMm[0−1/r]
U=−GMmr
Conservation of Energy
An object of mass m moving with velocity v1 at distance r1 from a massive body of mass M has the combined gravitational potential energy U and kinetic energy K:
U+K=−GMmr1+12mv21
If no energy is added into the system by using an engine, this total is conserved. At distance r2 and velocity v2 we will find the total is the same.
−GMmr1+12mv21=−GMmr2+12mv22
If we know the speed of an object at one distance, we can predict the velocity when the object is at a different distance.
Escape Velocity
Imagine you are standing on the surface of the Earth, and tossed a ball up. You will see the ball gradually slow down, briefly stopping in the air, and then come down again. If you throw the ball hard enough, the ball will slow down, and it will stop at infinity! This means the ball has completely escaped. The velocity that you throw the ball so that it escapes is called the escape velocity.
Imagine you are standing on the surface of the Earth, and tossed a ball up. You will see the ball gradually slow down, briefly stopping in the air, and then come down again. If you throw the ball hard enough, the ball will slow down, and it will stop at infinity! This means the ball has completely escaped. The velocity that you throw the ball so that it escapes is called the escape velocity.
When you are standing on the Earth, the surface is about 6371 km from the centre.
r1=6.37×106m
When the ball reaches infinity, the velocity is zero:
r2=∞
v2=0
We want to find the escape velocity, v1.
Using the principle of conservation of energy:
−GMmr1+12mv21=GMm∞+12m(0)2
12mv21=GMmr1
12v21=GMr1
v21=2GMr1
v1=√2GMr1
Putting in the numbers, with G=6.67×10−11m3kg−1s−2 and mass of Earth M=5.97×1024:
v1=√2×6.67×10−11×5.97×10246.37×106=1.09×104ms−1
That is 39,240 km/h.
What if you are already in space, at some distance away from the Earth? Let's say at a distance where r1=20000km? I invite you to try calculating it.
v1=√2×6.67×10−11×5.97×10246.37×106=1.09×104ms−1
That is 39,240 km/h.
What if you are already in space, at some distance away from the Earth? Let's say at a distance where r1=20000km? I invite you to try calculating it.
You can also try to calculate the escape velocity from the surface of other planets such as:
Planet | mass | radius |
---|---|---|
Mars | 6.39×1024kg | 3389.5 km |
Jupiter | 1.898×1027kg | 69911 km |
Asteroid Ceres | 9.38×1020 kg | 470 km |
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