- #1
James Brady
- 105
- 4
I'm trying to find out what would happen to a mass if it had a constant force applied to it for a very long time. It eventually approaches the speed of light, I want to plot it's velocity with respect to time as it gets harder and harder to push. Here's what I came up with.
From Newton's 2nd Law: dv/dt = F/M.
Making this relativistic: dv/dt = [itex]\frac{F}{M}[/itex] *[itex]\sqrt{1 - v^2/c^2}[/itex]
Setting up the differential equation:
[itex]\frac{dv}{\sqrt{1 - v^2/c^2}}[/itex] = [itex]\frac{F}{M}[/itex]*dt
I multiply both sides of the equation by 1/c to make it integrateable.
[itex]\frac{dv}{C * \sqrt{1 - v^2/c^2}}[/itex] = [itex]\frac{F}{C * M}[/itex]*dt
And solving for the integral of both sides I get...
sin^-1(v/c) = [itex]\frac{F * t}{M*C}[/itex]
v(t) = sin([itex]\frac{F * t}{M*C}[/itex]) * C
I'm pretty happy with this answer , the units check out alright, and it looks good. But once you get close to the speed of light, the velocity turns around and starts going back down because sine is one of those cyclic functions. I know of course you will not all of a sudden reverse acceleration once you near the speed of light though, so I know this isn't right.
Any help would much be appreciated, I will give you an e-high-five. Or an e-hug if you're into that sort of thing. Seriously, I'm that lonely.
From Newton's 2nd Law: dv/dt = F/M.
Making this relativistic: dv/dt = [itex]\frac{F}{M}[/itex] *[itex]\sqrt{1 - v^2/c^2}[/itex]
Setting up the differential equation:
[itex]\frac{dv}{\sqrt{1 - v^2/c^2}}[/itex] = [itex]\frac{F}{M}[/itex]*dt
I multiply both sides of the equation by 1/c to make it integrateable.
[itex]\frac{dv}{C * \sqrt{1 - v^2/c^2}}[/itex] = [itex]\frac{F}{C * M}[/itex]*dt
And solving for the integral of both sides I get...
sin^-1(v/c) = [itex]\frac{F * t}{M*C}[/itex]
v(t) = sin([itex]\frac{F * t}{M*C}[/itex]) * C
I'm pretty happy with this answer , the units check out alright, and it looks good. But once you get close to the speed of light, the velocity turns around and starts going back down because sine is one of those cyclic functions. I know of course you will not all of a sudden reverse acceleration once you near the speed of light though, so I know this isn't right.
Any help would much be appreciated, I will give you an e-high-five. Or an e-hug if you're into that sort of thing. Seriously, I'm that lonely.