What is the Mass of a Proton that is Travelling at 99.9% the Speed of Light?

Well… these terms are often used imprecisely. The mass of a proton or a rocket, or anything else, does not actually increase as it approaches the speed of light. Its rest-mass always stays the same, but its relativistic mass increases, as that’s a function of its rest mass and its speed relative to an observer.

The formula for working out the relativistic mass of an object moving in relation to your reference frame is:

Where:

m_r = relativistic mass (in kg)m_0 = rest mass (in kg)v = velocity (in meters/second)

c = speed of light (also in m/s)

The rest mass of a proton is roughly 1.67 x 10^-27 kg

The speed of light (in a vacuum) is 299,792,458 m/s

99.9% of the speed of light (in a vacuum) is 299,492,665.5 m/s

So our formula is:

This works out to:

So relativistic mass is 3.74 x 10^-26 kg

The relativistic mass of a proton (or anything) travelling at 99.9% the speed of light, is 22.4 x more than the rest mass of a proton (or whatever you’re measuring) when it’s in the same reference frame as yourself (not moving relative to you).

But Wait!

Just for fun….

The CERN’s Large Hadron Collider can speed particles up to 99.9999991% the speed of light (source) particle accelerator. So let’s see how much more massive a proton is, relativistically, when it is travelling at that ridiculous speed, compared to when it’s sitting down beside you, just chilling.

99.9999991% the speed of light (in a vacuum) is: 299,792,455.3 m/s (only about 6 miles per hour slower than the speed of light, which travels 186,000 miles per second!!); so:

Which gives us:

So relativistic mass is 1.246 x 10^-23

The relativistic mass of a proton (or anything) travelling at 99.9999991% the speed of light, is more than 7,000 times more than its rest mass!