Understand notation of motion equation in the context of differential equations.

Question

I'm working on practice problems to get familiar with differential equations, but can't quite understand the notation of the following problem:

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The motion equation $\dot v = g - \alpha v |v|$ is given for velocity v with acceleration $g > 0$ and quadratic friction of a sinking body with $\alpha > 0$ and initial value $(0) = v_0 \in \mathbb{R}$.

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Now it is asked to find out for which initial value $v$ will be constant. Since I have zero experience in physics and have just gotten started with differential equations, I'm struggling with the notation. I understand that $\dot v$ probably denotes $y^{\prime}$ and therefore $v$ will be $y$, but what is $g$? I would be very thankful for any clarifications.

Answer 1

There are many notations for differentiation: Leibniz's Notation $\frac {dy}{dx}$. Newton's dot notation $\dot{y}$. Euler's notation $Dy$. And the most common notation is surely Lagrange's notation $y'$. Alle these notations are equivalent. For g it's certainly a constant. And as Tanner wrote in the comment, $\dot v$ stands for $\frac {dv}{dt}$.

Now it is asked to find out for which initial value v will be constant?

When $v$ is constant, its derivative is $\dot v=0$

Answer 2

If you want to have a constant velocity you need $\frac {dv}{dt} =0$

That is $\alpha v|v| = g$

Solve for $v$ to get the initial velocity.