Solve relativistic differential force equation for velocity

You need to integrate both sides of this equation. Think about what the limits of integration are going to be.
  • #1
psal
4
0

Homework Statement


I proved that a relativistic 1D force is
F = [itex]\gamma[/itex]3*m*dVx/dt = m * dVx/dt * 1/ (1 - (v/c)2)3/2

Then, "This is a separable differential equation that can be solved using a trig
substitution. Use this (or some other technique that works) to show that the velocity is given by
v(t) = [itex]\frac{a*t}{\sqrt{1 + \frac{at}{c}2}}[/itex]

Homework Equations



a = [itex]\frac{dVx}{dt}[/itex] * [itex]\frac{1}{(1-\frac{v}{c}3/2}[/itex]
β = [itex]\frac{v}{c}[/itex] = sinΘ
cosΘ = [itex]\sqrt{1 - β2}[/itex]

The Attempt at a Solution


dβ = cosθdθ
a(t) = [itex]\frac{c*cosθdθ}{cos2θ}[/itex] = [itex]\frac{cdθ}{cosθ}[/itex]
I don't really know what to do from here to arive at the answer
 
Last edited:
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  • #2
psal said:

Homework Statement


I proved that a relativistic 1D force is
F = [itex]\gamma^3m\frac{dVx}{dt} = m \frac {dVx}{dt}\frac 1{(1 - (v/c)^2)^{3/2}}[/itex]

I think I fixed the Tex in your first equation. Don't use the sup and /sup tags in a tex expression. If you want an exponent of 3/2 just use ^{3/2} in the Tex. You also don't need the * for multiplication. You can use \cdot if you really want a multiplication sign. I will leave it to you to fix the rest if you desire. Also you can preview your posts before posting to see if the Tex is working.
 
Last edited:
  • #3
psal said:

Homework Statement


I proved that a relativistic 1D force is
$$F = \gamma^3 m \frac{dV_x}{dt} = m \frac{dV_x}{dt} \frac{1}{[1 - (v/c)^2]^{3/2}}.$$ Then, "This is a separable differential equation that can be solved using a trig substitution. Use this (or some other technique that works) to show that the velocity is given by
$$v(t) = \frac{at}{\sqrt{1 + \left(\frac{at}{c}\right)^2}}$$

Homework Equations


\begin{align*}
a &= \frac{dV_x}{dt} \frac{1}{[1 - (v/c)^2]^{3/2}} \\
\beta &= \frac{v}{c} = \sin \theta \\
\cos \theta &= \sqrt{1 - \beta^2}
\end{align*}

The Attempt at a Solution


$$d\beta = \cos\theta\,d\theta$$
$$a(t) = \frac{c\cos\theta\,d\theta}{\cos^2\theta} = \frac{c\,d\theta}{\cos\theta}$$
I don't really know what to do from here to arrive at the answer.
I take it you're using ##V_x## and ##v## to represent the same thing. Don't do that. Pick one. It also looks like you're supposed to assume the acceleration ##a## is constant.

After the substitution, you have
$$a\,dt = \frac{c \cos\theta \, d\theta}{(1-\sin^2\theta)^{3/2}}.$$ If you simplify that, you don't get what you got. Your last line, in particular, doesn't make sense. You shouldn't have a ##d\theta## all alone in the equation.
 

Related to Solve relativistic differential force equation for velocity

What is the relativistic differential force equation for velocity?

The relativistic differential force equation for velocity is given by F = dp/dt = d(mv)/dt = m(dv/dt) + v(dm/dt), where F is the force, p is the momentum, t is time, m is the mass, and v is the velocity. This equation takes into account the effects of relativity on the motion of an object.

How is this equation different from the classical force equation?

This equation is different from the classical force equation (F = ma) in that it takes into account the changing mass of an object as its velocity approaches the speed of light. The classical equation assumes a constant mass, while the relativistic equation considers the effects of mass increase at high velocities.

What is the significance of solving this equation?

Solving this equation allows us to accurately predict the motion of objects at high velocities, where classical mechanics fails to provide accurate results. It also helps us understand the effects of relativity on the behavior of particles in the universe.

What are some applications of this equation?

This equation is used in various fields such as astrophysics, particle physics, and nuclear engineering to study the behavior of particles at high speeds. It is also used in the development of technologies such as particle accelerators and space propulsion systems.

Are there any limitations to this equation?

Like any scientific equation, this equation has its limitations. It is only applicable to objects moving at high velocities, close to the speed of light. It also assumes a flat spacetime and does not account for phenomena such as gravity. Additionally, it cannot be used to describe the behavior of quantum particles.

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