Get time from acceleration given as a function of velocity

[harmyder]

Homework Statement

If the acceleration is given as a function of velocity a(v), we can found t(v) as t(v) = t0 + ∫^v_v0 1/a(v) dv.

Homework Equations

The Attempt at a Solution

I just can't understand there to go to understand the equation..

 

[gneill]

What is the (calculus based) definition of acceleration?

 

[harmyder]

$$\frac{dv}{dt} = a(t) \rightarrow v(t_1) = v(t_0) + \int_{t_0}^{t_1}a(t)dt$$
$$\frac{dt}{dv} = \frac{1}{a(t)} \rightarrow t(v_1) = t(v_0) + \int_{v_0}^{v_1}\frac{1}{a(t)}dv$$

but it is still not $a(v)$

 

[Nathanael]

$$t(v_1) = t(v_0) + \int_{v_0}^{v_1}\frac{1}{a(t)}dv$$

How will you integrate f(t)dv? t and v are not independent variables.

Note that acceleration a is dv/dt regardless of if a is a function of time a(t) or a function of velocity a(v)