### Question

What would be the force required to accelerate 1 gram to 20% of the speed of light?

Asked by:
Homer Connor

### Answer

It is not just a question of 'how much force' is needed, but rather a combination of a given
force for a given length of time. In other words, a small force for a long time can result
in the same velocity as a large force for a short time. This combination of force and time
is called IMPULSE, and equals the change in momentum given to any mass. Momentum is simply
mass x velocity.

20% of the speed of light is about 6 x 10^{7} meters/second. Since the relativistic effects at that velocity are small (only about 2%), let's ignore them and just find the impulse
needed in non-relativistic terms. A velocity increase given to 1 gm from 0 to 6 x 10^{7} m/sec means its momentum would have to change by: 0.001 kg x 6x10^{7} m/sec = 60,000 kg m/sec

So the IMPULSE needed is the equivalent of 60,000 kg m/sec. In the metric system, a NEWTON
is 1 kg m/sec^{2}, so any combination of newtons x seconds giving a product of 60,000 would do
the job. [The units of newtons x seconds = kg m/sec^{2} x sec = kg m/sec = momentum units]

A force of 60,000 Newtons for 1 second, for example, would provide the impulse needed, as
would a force of 1000 Netwons for 60 seconds.

Answered by:
Paul Walorski, B.A., Part-time Physics Instructor