1. Resistive (Drag) Forces
AP Physics
Mrs. Coyle

2. Falling with Air Resistance- Terminal Velocity

3. 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.

4. 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 = b v
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.

5. 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

6. 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

7. 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.

8. R = ½ DrAv2 Air Drag at High Speeds
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

9. Example 2: Find the expression for acceleration for a falling object with air resistance.
Ans: a= g-(D rA) v2 2m

10. For the same example find the terminal speed.
Ans:

11. Some Terminal Speeds