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Resistive (Drag)
Forces
AP Physics
Mrs. Coyle
Falling with Air Resistance- Terminal Velocity
Terminal Velocity
http://www.physicsclassroom.com/mmedia/newtlaws/efff.cfm
Resistive (Drag) Forces
• When an object moves through a medium (liquid or
gas), the medium exerts a resistive force, R, on the
object, opposing its direction of motion.
• The magnitude of the drag force depends on:
– the medium
– the size and shape of the object
– the speed of the object
• When equilibrium is reached between the drag force
and other forces acting on the object, then the
object travels with a terminal velocity.
Magnitude of the Resistive Force, R
R can depend on v in a variety of ways.
Here are two:
a)Slow motions or small objects
R=bv
b)Large objects
R = bv + cv2
Note:
b and c are constants, that depend on the
medium and the size and shape of the object.
Example 1:
Slow speed, small object
• Find the expression
for acceleration in
terms of v, b, g, m.
• Answer: a= dv = g- b v
dt
m
For the same example find the terminal
velocity, vt.
• a= dv = g- b v
dt
m
• When t=0 then v=0, so a= g
• When a=0, terminal velocity is reached:
a= 0= dv = g- b v 
vt= mg
dt
m
b
For the same example the general
expression for v is:
• v= vt (1-e-t/t )
• The time constant
t = m/b, is the
time by which the
sphere reaches
63.2% of its
terminal speed.
Air Drag at High Speeds
R=½
2
DrAv
– D : drag coefficient (for a sphere D=0.5)
– r : density of air
– A : cross-sectional area of the object
measured in a plane perpendicular to v
(for a sphere A= pr2 )
– v : speed of the object
Example 2: Find the expression for
acceleration for a falling object with air
resistance.
• Ans: a= g-(D rA) v2
2m
For the same example find the
terminal speed.
• Ans:
vT 
2mg
Dr A
Some Terminal Speeds