Answer:
The DE will be [tex]\frac{d^2x}{dt^2}-\frac{k}{m}\frac{dx}{dt}-g=0[/tex]
Explanation:
We have to find differential equation under the influence of gravity and experiencing a resistive force
Let an object of mass m falling under the influence of gravity
So the force experience under gravity [tex]=mg[/tex]
Le the a resistive force of magnitude kv opposes this gravity force, here k is constant and v is velocity.
So net force [tex]F_{NET}=mg-kv[/tex]-----eqn 1
[tex]F_{NET}=ma[/tex]
So [tex]ma=mg-kv[/tex]
We know that velocity is rate of change of position so [tex]v=\frac{dx}{dt}[/tex], and acceleration is rate of change of velocity so [tex]a=\frac{d^2x}{dt^2}[/tex]
Putting all these value in eqn 1
[tex]m\frac{d^2x}{dt^2}=mg-k\frac{dx}{dt}[/tex]
[tex]\frac{d^2x}{dt^2}-\frac{k}{m}\frac{dx}{dt}-g=0[/tex]
