An object mass m=1 is in free fall at a height h(t) of the surface. Considering the presence of a force of air resistance  `f_(r)=(dh)/(dt) ` and gravity g=9,8m/s^2, find the differential equation...

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Two forces will act on ball once it has been released (i) Garvitational Force (ii)The frictional force due to air resistance which work in opposite direction of motion. For ball is moving downword (i.e., when the velocity vector is positive), one of these forces are directed downward, and  opposite  direction of the motion of the ball.

The total force is the vector sum of the gravitational and downword l forces:
Force downword = DF
Gravitational Force= GF
Air resistance= AF
DF+GF=AF       (i)
Let velocity of the ball is  v m/s=dh/dt
so AF = `f_r`
GF=mg   (g acceleration due to gravity )
   =1 . 9.8
DF = m dv/dt    
( By Newton's second law of motion , dv/dt acceleration)
m dv/dt +mg=`f_r`
dv/dt+g-`(1/m)f_r` =0   ( since m=1)

This is applied when ball is moving downword .

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