Integrating velocity with respect to time when velocity depends on position

[timsea81]

Okay this seems like it should be simple but I'm stuck.

A particle moves from point A to point B in one dimension. The velocity of the particle changes with position according to some formula V = V(x). How long does it take for the particle to get from point A to point B?

If I could write V as a function of time, I could than integrate V=V(t) from 0 to t and that would give me the transit time. How do I do it if V depends on X?

Say, for example, V(X) = 1 + ax

 

[Xiao10]

dx=vdt, so dt=dx/v, integrate to find time

 

[timsea81]

Thanks. That does it, and I think I understand it now. I was thinking of it backwards before.

If I have v=v(t), I can integrate v from 0 to t to get the total distance traveled. I already know time in that case. If I don't have time I'd have to solve this for t, set it equal to t, and solve. In this case v = dx/dt --> dx = v dt, so integrating v dt gives you x.

If I have v=v(x) I solve so that dx is at the end of the expression: dt = (1/v) dx so I can integrate with respect to x, the variable that I know.

 

[rbj]

why can't you guys bother to use \LaTeX on this site that provides that resource?

it helps you articulate your question or answer. and it makes it easier for anyone to read either.

 

[timsea81]

Why can't you bother capitalizing the first words of your sentences, or proofreading your post to realize that the word "either" was used wrong?

I don't know latex, that's why I didn't use it. Do you similarly not know the English language?

I guess I might say "thanks for the tip" had you not worded your response as a criticism.