Projectile Motion(Two Dimensional)

Accounting for Drag

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.

20 2

1)sin()( gttVty

tVtx )cos()( 0

V0

Projectile Problem - No Drag

Velocity: Acceleration:Vx = Vocos() ax = 0

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

Position: y

x

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.

Projectile Problem - Drag

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

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)

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

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.

Drag Coefficient for Spheres

Projectile Problem - Drag Consider the

projectile, weighing W, and travelling at velocity V, at an angle .

Theta

W

VFD

+y

+x

The drag force acts opposite to the velocity vector, V.

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

+x

+y

DyF = F sin(theta)

DxF = F cos(theta)

D

D

Drag Forces The total drag force can be

computed by:

FD = 8.264 x 10-6 (CD V2 A)

where:|V2|= Vx

2 + Vy2

Drag Forces The X and Y components of the

drag force can be computed by:FDx = -FD cos()

FDy = -FD sin()

where: = arctan(Vy/Vx)

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 .

PAIRS EXERCISE 3 Develop an Excel spreadsheet that

describes the motion of a softball projectile:1) neglecting drag and 2) including drag

More

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

groundMore

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

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

Considerations for Pairs Exercise 3 What is a reasonable t ? What happens to the direction of

the drag force after the projectile reaches maximum height?

More

Sample Excel Spreadsheet

Sample Chart