Calculating Relativistic Velocity from Work and Mass

In summary, if you want to determine the relativistic velocity of a body starting from rest, you need to use conservation of energy and integrate the force over the distance.
  • #1
jaketodd
Gold Member
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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
 
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  • #2
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/physics/Relativity/SR/rocket.html [Broken]

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
 
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  • #3
Mentz114 said:
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/physics/Relativity/SR/rocket.html [Broken]

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
 
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  • #4
Use conservation of energy. The work done is equal to the increase in kinetic energy.
 
  • #5
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).
 
  • #6
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]
 
  • #7
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].
 
  • #8
It's getting over my head, but thanks all.

Jake
 
  • #9
DaleSpam said:
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.
 

1. How is relativistic velocity calculated from work and mass?

In order to calculate relativistic velocity, you can use the equation v = c * √(1 - (m0/m)^2), where v is the velocity, c is the speed of light, m0 is the rest mass, and m is the relativistic mass.

2. What units should be used for work and mass in the calculation?

Work is typically measured in joules (J) and mass is measured in kilograms (kg). However, it is important to make sure that both values are in the same units before plugging them into the equation.

3. Is this equation only applicable for objects moving at the speed of light?

No, this equation can be used for objects moving at any velocity, but it becomes more accurate as the velocity approaches the speed of light. For objects moving at slower speeds, the classical equation for kinetic energy (KE = 1/2 * mv^2) can be used.

4. Can relativistic velocity be negative?

Yes, relativistic velocity can be negative. This indicates that the object is moving in the opposite direction of the observer.

5. How does the relativistic mass affect the calculation of velocity?

As the relativistic mass increases, the velocity decreases. This is because the equation takes into account the increase in mass as an object approaches the speed of light, making it more difficult for the object to accelerate further.

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