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.