Projectile Motion (Two Dimensional)

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 Drag V0 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 = 0.00256 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. q Projectile Problem - Drag • Consider the projectile, weighing W, and travelling at velocity V, at an angle q. • The drag force acts opposite to the velocity vector, V.

11. q q 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

12. Drag Forces • The total drag force can be computed by: FD = 8.264 x 10-6 (CDV2 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 = 0.400 lbf • m = 0.400 lbm • Diameter = 3.80 in • Initial Velocity = 100 ft/s at 30o • CD = 0.6 • g = 32.174 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 = 32.174 (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