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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

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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.

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Projectile Problem - No Drag






Velocity: Acceleration:

Vx = Vocos(q) ax = 0

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

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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

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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

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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.

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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.

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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

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Drag Forces

  • The total drag force can be computed by:

    FD = 8.264 x 10-6 (CDV2 A)


    |V2|= Vx2 + Vy2

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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)

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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 .

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

    1) neglecting drag and

    2) including drag


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  • 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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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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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


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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?