1. Projectile Motion (Two Dimensional)
Accounting for Drag

2. Learning Objectives
Know the equation to compute the drag force on an object due to air friction
Apply Newton's Second Law and the relationship between acceleration, velocity and position to solve a two-dimensional projectile problem, including the affects of drag.
Prepare an Excel spreadsheet to implement solution to two-dimensional projectile with drag.

3. Projectile Problem - No DragV0 y
Position: q x
Velocity: Acceleration:
Vx = Vocos(q) ax = 0
Vy = Vosin(q) - g t ay = -g

4. Projectile Problem - Drag
All projectiles are subject to the effects of drag.
Drag caused by air is significant.
Drag is a function of the properties of the air (viscosity, density), projectile shape and projectile velocity.

5. General Drag Force
The drag FORCE acting on the projectile causes it to decelerate according to Newton's Law:
aD = FD/m
where: FD = drag force
m = mass of projectile

6. Drag Force Due to Air
The drag force due to wind (air) acting on an object can be found by:
FD = CDV2A
where: FD = drag force (lbf)
CD = drag coefficient (no units)
V = velocity of object (mph)
A = projected area (ft2)

7. Pairs Exercise 1
As a pair, take 3 minutes to convert the proportionality factor in the drag force equation on the previous slide if the
units of velocity are ft/s, and
the units of area are in2

8. Drag Coefficient: CD
The drag coefficient is a function of the shape of the object (see Table 10.4).
For a spherical shape the drag coefficient ranges from 0.1 to 300, depending upon Reynolds Number (see next slide).
For the projectile velocities studied in this course, drag coefficients from 0.6 to 1.2 are reasonable.

9. Drag Coefficient for Spheres

10. Projectile Problem - Drag
Consider the projectile, weighing W, and travelling at velocity V, at an angle q. q
The drag force acts opposite
to the velocity vector, V.

11. Projectile Problem - Drag
The three forces acting on the projectile are:
the weight of the projectile
the drag force in the x-direction
the drag force in the y-direction q q

12. Drag Forces The total drag force can be computed by:
FD = x 10-6 (CD V2 A)
where:
|V2|= Vx2 + Vy2

13. Drag Forces
The X and Y components of the drag force can be computed by:
FDx = -FD cos(q)
FDy = -FD sin(q)
where: q = arctan(Vy/Vx)

14. Pair Exercise 2 Derive equations for ax and ay from FDx and FDy.
Assuming ax and ay are constant during a brief instant of time, derive equations for Vx and Vy at time ti knowing Vx and Vy at time ti-1 .
Assuming Vx and Vy are constant during a brief instant of time, derive equations for x and y at time ti knowing x and y at time ti-1 .

15. PAIRS EXERCISE 3
Develop an Excel spreadsheet that describes the motion of a softball projectile:
1) neglecting drag and
2) including drag
More

16. PAIRS EXERCISE 3 (con’t)
Plot the trajectory of the softball (Y vs. X)
assuming no drag
assuming drag
Answer the following for each case:
max. height of ball
horizontal distance at impact with the ground
More

17. Data for Pairs Exercise 3
Assume the projectile is a softball with the following parameters:
W = lbf
m = lbm
Diameter = 3.80 in
Initial Velocity = 100 ft/s at 30o
CD = 0.6
g = ft/s2 (yes, assume you are on planet Earth)
More

18. Hints for Pairs Exercise 3
Reminder for the AES:
F = ma/gc
where gc = (lbm ft)/(lbf s2)
The equations of acceleration for this problem are:
ax = (FDx )gc/m
ay = (FDy -W)gc/m
More

19. Considerations for Pairs Exercise 3
What is a reasonable Dt ?
What happens to the direction of the drag force after the projectile reaches maximum height?
More

20. Sample Excel Spreadsheet

21. Sample Chart