# Relativity and terminal velocity

1. Nov 18, 2013

### mrcotton

Hi
If I had an object and a very long evacuated tube. Presumably the object with only the force of gravity acting on it would continuously accelerate and hence would have no terminal velocity.
However we know that it cannot go faster than light.
If my long tube is say pointing towards the centre of the Earth, how long would it have to be for an object to accelerate up to this speed.

If an object were to be released at infinity in a universe with only one planet would it reach the planet with a velocity equal to the escape velocity or would it have reached the speed of light.Any

thoughts to help my thoughts greatly appreciated
D

2. Nov 18, 2013

### ghwellsjr

That's true but it also cannot reach the speed of light.

Since it can never reach the speed of light, how about infinitely long?

If it's at infinity, then it will take forever to get to earth.

I hope you didn't think my suggestion that the tube be infinitely long was serious. The fact of the matter is that it is impossible.

3. Nov 18, 2013

### The_Duck

It would impact the planet with a speed equal to the planet's escape velocity. The way to see this is to imagine the situation in reverse: the object is launched upward from the planet's surface and eventually comes to rest "at infinity." The escape velocity is defined as the launch speed required for this to happen.

4. Nov 18, 2013

### ghwellsjr

5. Nov 18, 2013

### mrcotton

Hi George, and The_Duck, thanks for responding.
There is no need to disregard an answer they all make me think.
I was under the impreeion we needed a long tube infinetly long to bring a unit mass from infinity to define the joules per kilogram at that point in space.

So the velocity an object would reach a planet at is its escape velocity.
If we had an object with enough mass can we have a large enough force to accelerate an object up to the speed of light.

Would a black hole with an escape velocity of the speed of light attract an object from infinity to arrive at the speed of light?

Black holes have varying masses and if I remember correctly that the escape velocity is dependent on the amount of mass contained within a specific volume.

Thanks again, all responses help me in my own thinking processes

D

6. Nov 18, 2013

### yuiop

The terminal velocity for a test particle initially at rest at infinity, and free falling radially towards a non rotating spherical massive object, is given by:

$v = c \sqrt{\frac{2GM}{rc^2}}$

where v is the velocity of the particle measured by an stationary observer at r. If the massive body is not a black hole and the observer is at the surface of the body, then $r>2GM/c^2$ and the terminal velocity must always be less than c when the object impacts the surface. If the object is a black hole there can be no stationary observer at $r<=2GM/c^2$ so the terminal velocity still never exceeds c.