Answer to Question #104048 in Differential Equations for mm

Question

A particle of mass m is thrown vertically upward with velocity v0. The air resistance is mg cv^2 where c is a constant and v is the velocity at any time t. Show that the time taken by
the particle to reach the highest point is given by v0 underroot c= tan (gt underroot c )

Answer 1

A particle thrown vertically with air resistance.

Initial speed of a particle: $v_0$ (up). 

Air resistance modeled as $F_{air}=-cv^2,$ where $c$ is a constant and $v$ is the velocity at any time $t.$

Using Newton’s 2nd law gives:

"m \\text{\\"{y}}=-mg-c\\text{\\.y}^2"

Alternatively, one can write

$${dv \over dt}=-g-{c \over m}v^2$$

$$dt=-{dv \over g+{c \over m}v^2}$$

Integrate

$$\displaystyle\int_{0}^{t_h}dt=\displaystyle\int_{v_0}^0-{dv \over g+{c \over m}v^2}$$

$$t_h=-\sqrt{{m \over cg}}\bigg[\arctan\big(\sqrt{{c \over mg}}v\big)\bigg]\begin{matrix} 0 \\ v_0 \end{matrix}$$

$$t_h=\sqrt{{m \over cg}}\arctan\big(\sqrt{{c \over mg}}v_0\big)$$