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Projectile Motion (Two Dimensional) PowerPoint PPT Presentation

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Projectile Motion (Two Dimensional). Accounting for Drag. Learning Objectives. Know the equation to compute the drag force on an object due to air friction - PowerPoint PPT Presentation

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Projectile Motion (Two Dimensional)

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Projectile motion two dimensional l.jpg

Projectile Motion(Two Dimensional)

Accounting for Drag

Learning objectives l.jpg

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.

Projectile problem no drag l.jpg

Projectile Problem - No Drag






Velocity: Acceleration:

Vx = Vocos(q) ax = 0

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

Projectile problem drag l.jpg

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.

General drag force l.jpg

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 l.jpg

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)

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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 c d l.jpg

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 l.jpg

Drag Coefficient for Spheres

Projectile problem drag10 l.jpg


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.

Projectile problem drag11 l.jpg



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

Drag forces l.jpg

Drag Forces

  • The total drag force can be computed by:

    FD = 8.264 x 10-6 (CDV2 A)


    |V2|= Vx2 + Vy2

Drag forces13 l.jpg

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)

Pair exercise 2 l.jpg

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 l.jpg


  • Develop an Excel spreadsheet that describes the motion of a softball projectile:

    1) neglecting drag and

    2) including drag


Pairs exercise 3 con t l.jpg


  • 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


Data for pairs exercise 3 l.jpg

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)


Hints for pairs exercise 3 l.jpg

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


Considerations for pairs exercise 3 l.jpg

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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Sample Excel Spreadsheet

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Sample Chart

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