Slide 1 Projectile Motion(Two Dimensional)

Accounting for Drag

Slide 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.

Slide 3 ### Projectile Problem - No Drag

V0

y

Position:

q

x

Velocity: Acceleration:

Vx = Vocos(q) ax = 0

Vy = Vosin(q) - g t ay = -g

Slide 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.

Slide 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

Slide 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)

Slide 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

Slide 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.

Slide 9 ### Drag Coefficient for Spheres

Slide 10 q

### Projectile Problem - Drag

- Consider the projectile, weighing W, and travelling at velocity V, at an angle q.

Slide 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

Slide 12 Slide 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)

Slide 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 .

Slide 15 ### PAIRS EXERCISE 3

- Develop an Excel spreadsheet that describes the motion of a softball projectile:
1) neglecting drag and

2) including drag

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Slide 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

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Slide 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)

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Slide 18 ### Hints for Pairs Exercise 3

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Slide 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?

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Slide 20 ### Sample Excel Spreadsheet

Slide 21 ### Sample Chart