# Q & A: Energy, mass of particles, and speed

Q:
1. If an object (particle?) has no mass, how much energy does it take to increase its speed to C? 2. If this object had mass but it was extremely small, how much energy does it take to increase its speed to C?
- Tom Watkins (age 67)
Montpelier, VT, US
A:

Hello Tom3,

The short answer to question #1 is zero.

The short answer to question #2 is infinity.

Okay, why?

The relativistic relationship of mass, energy, and momentum is E2 = m2c4 + p2c2 and the relativistic velocity addition relation is  V2 = (V0+V1) /(1+V0V1/c2)  where V1 if the initial velocity, and V0 is the is the velocity added in its own rest frame.  If you plug in the value of c for the velocities of V0 or V1, the answer for V2 is still c.    A particle with zero rest mass starts out with velocity c so adding energy to it can't possibly increase its speed.   The  increase in its frequency of oscillation accounts for the extra energy.

The speed is just c*sqrt(1-m2c4/E2) so you can add all the energy you want to a particle with finite rest mass but you can never quite reach a velocity of c

LeeH

(published on 12/20/2013)

## Follow-Up #1: Reaching the speed of light and emitting light

Q:
Obviously a "what if" question but at issue is if the first particle or object started at rest. Then what energy would it take to get it to go C. I'm thinking that even a light beam would accelerate it to C. 2. Unrelated second question - If a star is receding from us at C would we ever see its light? What would be its red shift?
- Tom (age 67)
Montpelier, VT, US
A:

Hi Tom,

Experiments have been done where more and more energy was pumped into electrons to see how fast they would go. Classically, you would expect them to keep speeding up, but relativistically, they just approach the speed of light asymptotically. They never reach c, because they have mass.

As you near the speed of light, the amount of energy you need to apply to speed the particle up even a tiny bit explodes towards infinity.

2. Stars can't travel at the speed of light, so we can always see light which they emit. The redshift increases as the speed increases. I don't know of anything which can travel at the speed of light and still emit photons, so that kind of removes the craziness you may have been hoping for.

Cheers,

David Schmid

p.s. In the broad framework of a universe with accelerating expansion, a star which had once been visible can recede from us faster than c. However, it then becomes unobservable by any means. It's outside our cosmic horizon. Nothing it does has any effect here. /mw

(published on 12/22/2013)