- #1
samoth
- 8
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This is an extra-credit problem. The question covers materials some students may not have had prior, hence not being mandatory.
Question posed:
Force is given by
F=dp/dt
F=d/dt (mv)
F=m dv/dt
F=ma
and a=F/m is the slope of the line on a v versus t graph. This does not take into account the 'speed limit' of c.
Where
F=m dv/dt
dv=a dt
[tex]\int_{0}^{v} dv = \int_{t=o}^{t} a dt[/tex]
[tex]v = a \int_{t=o}^{t} dt[/tex]
v=at
**I am not sure the significance of v=at, but it was given**
So taking into account p=mv[tex]\gamma[/tex]
We have
F=dp/dt
F=d/dt (mv[tex]\gamma[/tex])
F=[tex]\gamma[/tex]m dv/dt
(F/m) dt=[tex]\gamma[/tex] dv
[tex]\gamma[/tex] dv=a dt
**then when integrating those terms, I have a problem with...**
[tex]\int_{0}^{v} \frac{dv}{\sqrt{1 - v^2/c^2}}[/tex]
...on the left side of the equation.
I have only had two semesters of calculus, so I am unsure if I am simply forgetting something, of if there is something here I don't completely know. I was told by a classmate that doing this involves differential equations, of which I have little knowledge.
The problem is to find the function that shows the line that asymptotically approaches c. I believe it involves a exponential, which is where I am faltering on my calculus.
Thanks to anyone who can help/explain or point me in the right direction!
***apologies if LaTeX does not work on the first posting***
Question posed:
Force is given by
F=dp/dt
F=d/dt (mv)
F=m dv/dt
F=ma
and a=F/m is the slope of the line on a v versus t graph. This does not take into account the 'speed limit' of c.
Where
F=m dv/dt
dv=a dt
[tex]\int_{0}^{v} dv = \int_{t=o}^{t} a dt[/tex]
[tex]v = a \int_{t=o}^{t} dt[/tex]
v=at
**I am not sure the significance of v=at, but it was given**
So taking into account p=mv[tex]\gamma[/tex]
We have
F=dp/dt
F=d/dt (mv[tex]\gamma[/tex])
F=[tex]\gamma[/tex]m dv/dt
(F/m) dt=[tex]\gamma[/tex] dv
[tex]\gamma[/tex] dv=a dt
**then when integrating those terms, I have a problem with...**
[tex]\int_{0}^{v} \frac{dv}{\sqrt{1 - v^2/c^2}}[/tex]
...on the left side of the equation.
I have only had two semesters of calculus, so I am unsure if I am simply forgetting something, of if there is something here I don't completely know. I was told by a classmate that doing this involves differential equations, of which I have little knowledge.
The problem is to find the function that shows the line that asymptotically approaches c. I believe it involves a exponential, which is where I am faltering on my calculus.
Thanks to anyone who can help/explain or point me in the right direction!
***apologies if LaTeX does not work on the first posting***
Last edited: