# Adding Many Velocities in Relativity

*Most recent answer: 06/19/2013*

Q:

An object is moving with some velocity with respect to a fixed point and a second object is thrown from the surfac of the first object such that the velocity of the second object with respect to the fixed point is twice as much as that of the first object. What will happen if this process of throwing objects is repeated for an infinit number of times.

- Anupam kumar (age 16)

Patna, Bihar, India

- Anupam kumar (age 16)

Patna, Bihar, India

A:

As the number of throws goes to infinity, the velocity as seen by the point you call "fixed" approaches c, the speed of light.

It's not hard to solve the continuous form of this problem, with lots of throws (N) each adding just a little v, where v is a small fraction of c. Then after N throws the velocity of the ball in the "fixed" reference frame will be c*tanh(Nv/c), where "tanh" is the hyperbolic tangent function.

Here's a picture of how the velocity will change as a function of Nv/c.

Mike W.

*(published on 06/19/2013)*