# F = ma doesn't work when aproaching c

1. Aug 10, 2010

### Jacob_hull

Hello all,

Im only a student, taking physics in college in september, However I have read up on the level of physics I will be studying. One thing that I couldn't get my head round, Why when an object aproaches c does its mass increase? Am I right in saying this also makes it impossible from the object to reach c? Also why does this cancel out f = ma?

Sorry if this seems like a stupid question, but it just sparked an interest in my mind and I couldn't find an explanation that actually fully explained it to me. Thanks all.

2. Aug 10, 2010

### bcrowell

Staff Emeritus
Welcome to Physics Forums!

There are basically two ways of describing this in relativity.

The more old-fashioned way is to write F=ma, and say that $m=\gamma m_o$, where $m_o$ is the mass the object has when it's at rest.

More recently (roughly within the last 50 years), it's been more common to write $F=m\gamma a$, and treat m as a constant.

So depending on which description you prefer, you can either say that mass increases or it doesn't increase, but either way the prediction is the same.

It doesn't cancel out F=ma completely. At velocities that are small compared to the speed of light, $\gamma$ is very close to 1, so F=ma is approximately right. That's why people believed F=ma was exact for hundreds of years.

Yes, this is one way of seeing that an object can't reach c. As its velocity gets close to c, $\gamma$ approaches infinity, so the force required in order to maintain the same acceleration also approaches infinity.

For an explanation of why this happens: http://www.lightandmatter.com/html_books/6mr/ch01/ch01.html#Section1.3 [Broken]

Last edited by a moderator: May 4, 2017
3. Aug 10, 2010

### LostConjugate

The mass increases because m = e/c^2 and E = 1/2mv^2

This means that if your velocity increases your energy increases your mass increases.

You are correct, a massive object can't reach the speed c.

Also f=ma does not work at speed c because the force would be infinite along with the mass which essentially states 1=1 since "a" now equals next to 0. Perhaps it works it is just not very interesting.

4. Aug 12, 2010

### Jacob_hull

Thanks for the reply, this sort of stuff doesn't get explained to you in Secondary school it was basically f=ma end of. But I understand now so thanks alot :)

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