## Wednesday, September 21, 2011

### Escape Velocity take 2

Currently re-reading "Time Travellers in Einstein's Universe" by J Richard Gott and reached a section on cosmic strings that touched on black holes and thought "That's not right". He was mentioning escape velocity and stated that the astronauts who went to the moon had to achieve this speed. No they didn't.

I have already covered it here, but another take won't hurt.

Imagine I throw a ball in the air so that it leaves my hand at a velocity of 30m/s upwards. Gravity pulls it down at 10m/s/s. After one second my ball is travelling at 20m/s; after two seconds 10m/s; after three seconds 0m/s. Then it starts to fall. Plotting a graph we get a diagonal line starting at 30m/s and sloping down and to the right until it reaches 0m/s after three seconds. The distance travelled upwards by the ball is the area of this triangle (30*3/2)=45m.

What if there was a barrier at a height of 25m which the ball could pass through but gravity wouldn't? After one second my ball has travelled 25m (20+((30-20)/2)) so it's passed through the barrier and is no longer being pulled back by gravity. As it's travelling at 20m/s with no restraint it will just keep travelling at 20m/s.

Now what if I'd only thrown it upwards at 20m/s? After one second it'll have travelled 15m and be at a velocity of 10m/s upwards. After two seconds it'll have travelled another 5m and be at 0m/s.Having travelled for only 20m it won't have reached the barrier and so is still subject to gravity and begins to fall.

So we can see that in order for the ball to reach the barrier it needs to be travelling somewhere between 20 and 30m/s that exact figure is the escape velocity.

Instead of a ball let's consider a rocket. This is a special rocket in that it has an infinite fuel source, but the design of the engine means it can only provide an accelerating thrust of 10m/s/s. While it's sitting on the ground I can provide an external thrust to get it into the air, but due to its mass I can only manage to throw it at 1m/s.

As 1m/s is no-where near out escape velocity of between 20 and 30m/s there's no way this rocket will escape the pull of gravity, but let's plot it out anyway.

After one second our rocket is subject to gravity at 10m/s/s downwards, but my engine is provided a force of 10m/s/s upwards. These cancel out. As there's no net force acting on the rocket it continues at its original velocity of 1m/s upwards. It has now travelled 1 metre upward. After two seconds exactly the same thing; it's still at 1m/s and has travelled another 1 metre upward.

In fact this will continue until after 25 seconds it reaches the barrier having travelled the full 25m upwards. After that without gravity the only force will be the engine and the rocket will start to accelerate at 10m/s/s.

So our rocket managed to break away from gravity and did so at a constant speed of 1m/s.

Back to my book and it should be seen now that the astronauts didn't need to reach escape velocity because they had a continuous upwards accelerative force acting on them.

To continue with the book black holes are so called because their escape velocity is greater than that of the speed of light meaning nothing can escape their grasp. Except hopefully you'll see that's not the whole picture. What is required is an acceleration that is (at least in the initial stages) slightly higher than that of the gravity of the black hole. Provided that a rocket can accelerate faster than the black hole's gravity and can maintain a counter-force equal to the black hole's pull there's no reason it can't leave due to not reaching escape velocity.

There are other reasons why escape is non-feasible, but escape velocity isn't one of them.