Relativistic version of Newton's second law with parallel force

In summary, the conversation discusses how to show that F=γ3ma when the force is parallel to velocity, using the equations F=dp/dt and p=γmv. The solution involves taking the derivative of p with respect to time and correctly applying the chain rule for the derivative of γ. This results in the equation F=dp/dt=γma[γ2(v/c)2+1], which can be simplified to F=γ3ma by setting (v/c)2 equal to 1-1/γ2. The conversation concludes with a thank you to the contributor for their help.
  • #1
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Homework Statement


"Given F=dp/dt. If the force is parallel to velocity show that F=γ3ma."


Homework Equations


F=dp/dt and p=γmv


The Attempt at a Solution


Since the force is the first derivative of the momentum with respect to time, and γ and v both vary with time since there is a net force causing both quantities to change, take the derivative of p using multiplication rule: dp/dt = m[γ'v+γa] γ'=γ3*v/c2 so this would change dp/dt to dp/dt = m[γ3(v/c)2+γa] After this I can't think of anywhere to go that could simplify this to the desired equation F=γ3ma.
 
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  • #2
are you sure about [tex]\gamma[/tex] and its derivative? there must be something wrong about it
 
  • #3
Yeah, I did the derivative wrong, I forgot to do chain rule for the v2 so the derivative of γ would be γ3*va/c2

So then dp/dt = γma[γ2(v/c)2+1]. Assuming the math I did was right the only way I could get what I'm looking for is if (v/c)2 is equal to 1-1/γ2 so when you distribute you'd get γ2-1+1, then it would give dp/dt = F = γ3ma what I'm looking for.

I'll have to mess around with this.

[edit]
Coincidentally (v/c)2 does equal what it needs to equal to simplify to what the equation needs to be.

Thank you a lot Pull :)
[/edit]
 
Last edited:

1. What is the relativistic version of Newton's second law with parallel force?

The relativistic version of Newton's second law with parallel force is a modification of the classical Newton's second law, which takes into account the effects of special relativity. It states that the force acting on an object is equal to the rate of change of its momentum, taking into account the relativistic mass and velocity of the object.

2. How does the relativistic version of Newton's second law differ from the classical version?

The main difference between the relativistic version of Newton's second law and the classical version is the inclusion of relativistic effects. In the classical version, the mass and velocity of an object are assumed to be constant, while in the relativistic version, the mass is allowed to vary with the object's velocity, and the velocity is considered to be a fraction of the speed of light.

3. Why is the relativistic version of Newton's second law important?

The relativistic version of Newton's second law is important because it allows us to accurately describe the motion of objects with high velocities, which cannot be explained by classical mechanics. It also helps us to understand the relationship between mass, energy, and momentum in special relativity.

4. How is the relativistic version of Newton's second law derived?

The relativistic version of Newton's second law can be derived from the principles of special relativity, specifically the Lorentz transformation equations. These equations relate the measurements of length, time, and velocity between two inertial reference frames, and they provide the basis for understanding how the mass and velocity of an object change with respect to an observer's frame of reference.

5. Does the relativistic version of Newton's second law apply to all objects?

Yes, the relativistic version of Newton's second law applies to all objects, regardless of their mass or velocity. However, its effects become more significant as the velocity of an object approaches the speed of light. For objects with low velocities, the relativistic version reduces to the classical version of Newton's second law.

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