What Goes Up Must Come Down – Sometimes

You throw a ball in the air. The harder you throw it the higher it goes and the longer it remains in the air. Is it possible to throw it so hard that it goes up but never comes down?

Well to do this you have to throw it so hard that its kinetic energy

(where m is the mass of the object and v is its velocity)

is greater than the energy of the Earth’s gravitational pull1

(where G is Newton’s gravitational constant, M is the mass of the Earth and r is the radius of the Earth)

So we can now work out at what point these two are equal:

divide both sides by m
multiply both sides by 2
take the square root of both sides

We now have an expression for the escape velocity i.e. the velocity that something has to be traveling so that it can escape the gravitational pull of the Earth.

Now we can put some numbers in. Newtons gravitational constant G = 6.674×10−11 N·kg–2·m2, the mass of the Earth M = 5.972 × 10²⁴ kg and the radius of the Earth r =6371km = 6.371 x 106m.

This gives a figure of 11,186m/s. If an object is thrown faster than this it would go up and never come down.

Fast But Not That Fast

11,186m/s is very fast – about 10 times the speed of sound. The Sun has an even higher escape velocity 617,500m/s. Is it possible to have an object so dense that even light cannot escape? A Black Hole.

Well the speed of light (c) is 3×108m/s. We can put this into the equation that we have just calculated instead of v:

replace velocity v with speed of light c
square both sides of the equation
multiply both sides of the equation by r
divide both sides of the equation by c2

So we can now work out what the radius of an object of mass M has to be for it to be a black hole. Let us take the example of an object with the same mass as the Earth. If we do the calculation it comes out as 0.00886m which is just under 1cm.

Therefore if the mass of the Earth was compressed into a sphere just under 1cm in radius it would be a black hole.


1 Note that the energy is not the same as the force which is given by Newton’s equation


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