The velocity of an object: $\frac{dv}{dt} = -kv^n$, k > 0, n constant. [Engineering Differential Equations Question]

Question

The velocity of an object: $\frac{dv}{dt} = -kv^n$, k > 0, n constant. At t = 0, the object begins moving with an initial, positive velocity of vₒ, and continues until its velocity reaches zero (if it ever does).

1. If 1 ≤ n < 2, show that the maximum distance travelled by the object in a finite time is less than $\frac{(vₒ^{2-n})}{(2-n)k}$.

2.Show that if n ≥ 2, then there is no limit to the distance the object can travel.

Answer 1

Hint:

Why do you not solve the equation: $$\frac{dv}{v^n}=-kdt$$? For your Control: the general solution is given by $$v(t)=((-1+n)(kt-C))^{1/(1-n)}$$