Applying Forces to Particles: Questions Answered

[res3210]

Hello everyone,

I am not completely familiar with the way small particles behave, but I assume that if one applies an acceleration to a particle(such as an electron) then that particle will accelerate. So here is my hypothetical: suppose we apply an acceleration of 1 m/s^2 to an electron. After about 300000000 seconds, it should be traveling at the speed of light. However, we know no physical particle can reach c, so that would mean that the constant acceleration would have to in fact decrease over time. Is this the case? And if so, what would happen if we applied an increasing acceleration to a particle? Also, would this mean that the constant acceleration would never reach zero, and we would be looking at a series which only achieves a value of c with respect to velocity when t equals infinity?

 

[Mentz114]

The theory of particles under constant acceleration has been well studied. See for instance

http://en.wikipedia.org/wiki/Rindler_coordinates

or

http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html

As you suggest, the speed of light becomes an asymptotal value.

 

[BruceW]

yeah, and keep in mind the difference between 1) constant proper acceleration and 2) constant acceleration, according to some inertial observer. As you have implied, it is not possible for a particle to be in constant acceleration over an arbitrary amount of time. The acceleration must eventually tend to zero. (and here I am talking about the 2nd definition of acceleration).

 

[Naty1]

As you imply, at high velocities speed and acceleration are not linearly related but related by hyperbolic [asymptotic] relationships..

There is a straightforward diagram here:

http://en.wikipedia.org/wiki/Proper_acceleration#Acceleration_in_.281.2B1.29D

Based on a quick scan of the article explanations, I'd look elsewhere for more detailed reading.

edit: You might find the relationship to the Lorentz factor of interest...it's a hyperbolic function as well. This reflects the actual law of velocity addition at higher speeds...

It's not the simple S = V + U we all learned in grade school...

http://en.wikipedia.org/wiki/Velocity-addition_formula#Special_theory_of_relativity

 

[Agerhell]

Hello everyone,

I am not completely familiar with the way small particles behave, but I assume that if one applies an acceleration to a particle(such as an electron) then that particle will accelerate. So here is my hypothetical: suppose we apply an acceleration of 1 m/s^2 to an electron. After about 300000000 seconds, it should be traveling at the speed of light. However, we know no physical particle can reach c, so that would mean that the constant acceleration would have to in fact decrease over time. Is this the case? And if so, what would happen if we applied an increasing acceleration to a particle? Also, would this mean that the constant acceleration would never reach zero, and we would be looking at a series which only achieves a value of c with respect to velocity when t equals infinity?

If you accelerate an electron in a constant electric field you will not get a constant acceleration, but you will get a constant increase in momentum according to:

$$\frac{d(m\bar{v})}{dt}=q(\bar{E}+\bar{v}\times\bar{B})$$

What you do in physics is not that you "apply an acceleration", instead you apply a force. The problem is that in electromagnetics the mass of the electron ("m" above) can be considered as varying with the velocity according to the "Lorentz factor". This means that in order to reach "c" you would have to have an infinitely strong force, which is not possible. A constant force does not result in a constant acceleration (which it does in classical physics), but it does result in a constant increase of momenta.

 

[res3210]

So then you are saying we observe a uniform change in motion?

 

[Naty1]

So then you are saying we observe a uniform change in motion?

nope...

If you accelerate an electron in a constant electric field you will not get a constant acceleration...

Mentz's links or mine illustrate all that.