Determining relativistic velocity from work done from rest with a given rest mass


by jaketodd
Tags: determining, mass, relativistic, rest, velocity, work
jaketodd
jaketodd is offline
#1
Feb25-12, 11:04 AM
PF Gold
P: 298
I am wondering how to determine relativistic velocity from a given amount of work done on a body starting from rest with a given rest mass.

This is not homework.

Thanks in advance,

Jake
Phys.Org News Partner Science news on Phys.org
NASA's space station Robonaut finally getting legs
Free the seed: OSSI nurtures growing plants without patent barriers
Going nuts? Turkey looks to pistachios to heat new eco-city
Mentz114
Mentz114 is offline
#2
Feb25-12, 11:32 AM
PF Gold
P: 4,081
It depends where the work is being done. If the accelerating body is a rocket then the relativistic rocket equations apply. See for instance

http://math.ucr.edu/home/baez/physic...SR/rocket.html

If the work is being done from outside the accelerated body ( such as in a particle accelerator ) the relativistic expression is the same as the Newtonian one but for a factor of γ or γ2
jaketodd
jaketodd is offline
#3
Feb25-12, 11:58 AM
PF Gold
P: 298
Quote Quote by Mentz114 View Post
It depends where the work is being done. If the accelerating body is a rocket then the relativistic rocket equations apply. See for instance

http://math.ucr.edu/home/baez/physic...SR/rocket.html

If the work is being done from outside the accelerated body ( such as in a particle accelerator ) the relativistic expression is the same as the Newtonian one but for a factor of γ or γ2
Thanks for the link, but I still could use some help...

Here's the equation it provides for velocity:
v = at / sqrt[1 + (at/c)2]

How do we get 'a' and 't' from a given amount of work done on a given mass from rest?

Maybe I shouldn't have said "relativistic velocity." I just want to know how to get the velocity it has compared to when it was at rest, due to the given amount of work done to the given mass.

Thanks,

Jake

Mentz114
Mentz114 is offline
#4
Feb25-12, 12:44 PM
PF Gold
P: 4,081

Determining relativistic velocity from work done from rest with a given rest mass


Use conservation of energy. The work done is equal to the increase in kinetic energy.
elfmotat
elfmotat is offline
#5
Feb25-12, 01:30 PM
elfmotat's Avatar
P: 260
You could set up an integral of the force dp/dt over dx, where p is the relativistic momentum. This would be unnecessary work, however, because when you evaluate it you just end up with the relativistic kinetic energy formula (as per Mentz's suggestion).
DaleSpam
DaleSpam is online now
#6
Feb25-12, 03:25 PM
Mentor
P: 16,477
The total energy is given by
[tex]\frac{mc^2}{\sqrt{1-v^2/c^2}}[/tex]

It is also given by
[tex]mc^2+w[/tex]

Set those two expressions equal and solve for v which gives:
[tex]v=\pm\frac{c \sqrt{w \left(2 c^2 m+w\right)}}{c^2 m+w}[/tex]
robphy
robphy is offline
#7
Feb25-12, 08:32 PM
Sci Advisor
HW Helper
PF Gold
robphy's Avatar
P: 4,108
In terms of rapidities,
DaleSpam's expressions would be
[itex]mc^2 \cosh\theta[/itex] and [itex] mc^2+w [/itex],
where velocity [itex]v=\tanh\theta[/itex].

So, [itex]v=\tanh\left(\cosh^{-1}\left(1+\displaystyle\frac{w}{mc^2}\right)\right)[/itex].
jaketodd
jaketodd is offline
#8
Feb26-12, 07:57 PM
PF Gold
P: 298
It's getting over my head, but thanks all.

Jake
yuiop
yuiop is offline
#9
Feb27-12, 07:27 PM
P: 3,967
Quote Quote by DaleSpam View Post
Set those two expressions equal and solve for v which gives:
[tex]v=\pm\frac{c \sqrt{w \left(2 c^2 m+w\right)}}{c^2 m+w}[/tex]
If we rewrite Dalespam's expression slightly differently as:

[tex]v=\pm c \sqrt{1- \left(\frac{mc^2}{ mc^2+w}\right)^2}[/tex]

it is slightly easier to see that the work required to achieve v=c is infinite.


Register to reply

Related Discussions
Determining Force applied so mass will stay at rest Introductory Physics Homework 3
Mass (both relativistic and rest) of a photon Special & General Relativity 16
Re: rest mass/relativistic mass question General Physics 3
Re: rest mass/relativistic mass question General Physics 0
Relativistic and rest mass Special & General Relativity 10