Consider a body with mass m in free fall with air resistance depending quadratically of speed. If h=h(t) denotes the distance from the object to the ground, g is the acceleration of gravity...

Consider a body with mass m in free fall with air resistance depending quadratically of speed. If h=h(t) denotes the distance from the object to the ground, g is the acceleration of gravity and β>0 is a coefficient of resistance, determine the differential equation that models the problem, regardless of the object being downward or upwards.

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pramodpandey | College Teacher | (Level 3) Valedictorian

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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 upward (i.e., when the velocity vector is positive), both 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 frictional forces:
Force upword = UF
Gravitational Force= GF
Air resistance= AF
 
UF=GF+AF       (i)
 
Let velocity of the ball is  v m/s=dh/dt
 
so AF = (beta)v^2
 
GF=mg   (g acceleration due to gravity )
 
UF = m dv/dt    
 
( By Newton's second law of motion , dv/dt acceleration)
 
m dv/dt= -mg-(beta) v^2
 
dv/dt+(beta/m)v^2+g=0
 
(d^2h)/dt^2+(beta/m)(dh/dt)^2+g=0

This is applied when ball is moving upword .
In case ball moving down word equation become.

UF+GF = AF

and the differential equation describing the descending portion of the ball's trajectory is:

(d^2h)/dt^2-(beta/m)(dh/dt)^2+g=0
 

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