velocity from acceleration if acceleration is a function of space...

fisico30

Hello Forum,

usually the acceleration a is a function of time: a(t)= d v(t) /dt

to find v(t) se simply integrate v(t)= integral a(t) dt

What if the acceleration was a function of space, i.e. a(x)?

what would we get by doing integral a(x) dx? The velocity as a function of space, v(x)?

but a(x) is not defined to be v(x)/dx or is it? maybe some chain rule is involved..

thanks,

fisico30

rcgldr

You use chain rule

a(x) = dv/dt = f(x)

multiply dv/dt by dx/dx:

(dv/dt)(dx/dx) = (dx/dt)(dv/dx) = v dv/dx = f(x)

multiply both sides by dx

v dv = f(x) dx

If f(x) can be integrated and the integral of f(x) = g(x), then

1/2 v² = g(x) + c

This results in an equation that relates velocity and position.

v = sqrt(2 (g(x) + c))

To get time versus position, you start with

v = dx/dt = sqrt(2 (g(x) + c))

and then integrate

dx/(sqrt(2 (g(x) + c))) = dt

assuming h(x) is the integral of dx/(sqrt(2 (g(x) + c))), you get

t = h(x) + d

where d is constant of integration. It may be possible to solve this equation to get x = some function of t.

One simple case is a(x) = -x, which results in x(t) = e(sin(t+f)), where e and f are constants.

fisico30

Thanks rgcldr!

great explanation and example!

fisico30

fisico30

Hi rcgldr,

one question:

when you get v dv = f(x) dx, is v a function of t or of x, i.e. is v equal to v(t) or v(x)?

It looks like it would be a function of x, v(x), since 1/2 v^2 = g(x) + c results in v(x) = sqrt(2 (g(x) + c)).

I am confused because when we write v=dx/dt I always assume that v must be a function of time t, i.e. v(t).
When in your first step you write a(x) = dv/dt I tend to think that the v must be function of t since dv/dt is a derivative with respect to time.

So which one is correct? a(x)= dv(x)/dt or a(t)=dv(t)/dt. Acceleration is always the time derivative of the velocity vector but the velocity vector can be a function of any dependent variable: v(t), v(x), v(F), where F is force, etc....

Thanks,
fisico30

fisico30

I guess my point is: if a function, like a, is defined as a time derivative of another function, dv/dt, does the differentiated function need to be a function if time?

rcgldr

At this point, it's just a relationship betwen v and x.

It is, but I was only able to take advantage of that in the last step where I find t as a function of x.

True, but I used chain rule to get rid of the dt and end up with v dv = f(x) dx.

Both are correct, but if acceleration is defined as a function of x, then it may not be possible to find an equation for acceleration as a function of time. In the simple example I gave, a(x) = -x, you will be able to find both a(x) and a(t). I ended up finding x(t) = e sin(t + f). You can take the derivative of this to find v(t), and the derivative of v(t) to find a(t), so a(x) = -x, and a(t) = -e sin(t+f). For other situations, you may not be able to solve for x(t), v(t), or a(t), in which case numerical integration will be required.

fisico30

So it is mathematically legal and ok to write a derivative of a function even if
the independent variable a of the function and the differentiation variable b are not the same: f(a) and db to create df(a)/db...

I guess that is all the chain rule is about....