Integrate acceleration when a = f(v)

[Fuergrissa]

Mentor note: Moved from a technical forum section, so missing the HW template.
Summary:: Integrate acceleration when a = f(v) when separation of variables is not trivial, ie a = k +v^2

When acceleration is a function of velocity, ie there is a friction force, you would separate the variables as such:

dv/dt = v^2

v^-2*dv = dt

and then integrate.
but what do you do when they are not easily separable, as in:

dv/dt = k + v^2

?

 

[PeroK]

Isn't $\frac 1{k + v^2}$ a standard integral?

 

[hunt_mat]

Note that
$$a=\frac{dv}{dt}=\frac{dv}{dx}\cdot\frac{dx}{dt}=v\frac{dv}{dx}$$
Or you can just separate as usual to get:
$$\frac{1}{k+v^{2}}\frac{dv}{dt}=1$$
Integrate both sides to get:
$$\int\frac{dv}{k+v^{2}}=\int dt+C$$