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Friday, December 25, 2020

Gravity (Part 1)

Newton's Law of Gravitation is given by:

F=GMmr2

Where M and m are the masses of the two objects, r is the distance between them, and  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=rGMmr2dr

U=GMmrr2dr

U=GMm[r1]r

U=GMm[1r]r

U=GMm[01/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.

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=6.67×1011m3kg1s2 and mass of Earth M=5.97×1024:

v1=2×6.67×1011×5.97×10246.37×106=1.09×104ms1

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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