Resistance is Futile Factoring Air Resistance into Projectile Motion - - PowerPoint PPT Presentation

Resistance is Futile

Resistance is Futile

Factoring Air Resistance into Projectile Motion Joseph Hays May 16, 2014

Resistance is Futile

Introduction

The goal of this presentation is to explain how to set up, solve, and compare two commonly used models of projectile motion through two dimensions: a model which assumes negligible air resistance, and a model which assumes linear air resistance.

Resistance is Futile

Linear Resistance

Fdrag = bV As the term implies, the linear model fairly assumes that the force

• f drag is directly proportional to projectile velocity - a model that

usually works well at low velocity, where viscous drag dominates.

Resistance is Futile

Negligible Air Resistance

Fdrag = 0 This model is usually presented in introductory physics classes and derived with kinematics equations.

Resistance is Futile

Projectile Motion through Two Dimensions

When describing projectile motion through two dimensions, two components of motion must be considered: the horizontal component of motion and the vertical component.

Resistance is Futile

Strategy for Deriving Trajectory

1) Analyze the horizontal and vertical components of motion separately.

Resistance is Futile

Strategy for Deriving Trajectory

1) Analyze the horizontal and vertical components of motion separately. 2) Derive an equation for horizontal displacement as a function of time.

Resistance is Futile

Strategy for Deriving Trajectory

1) Analyze the horizontal and vertical components of motion separately. 2) Derive an equation for horizontal displacement as a function of time. 3) Derive an equation for vertical displacement as a function of time.

Resistance is Futile

Strategy for Deriving Trajectory

1) Analyze the horizontal and vertical components of motion separately. 2) Derive an equation for horizontal displacement as a function of time. 3) Derive an equation for vertical displacement as a function of time. 4) Solve the horizontal displacement for time.

Resistance is Futile

Strategy for Deriving Trajectory

1) Analyze the horizontal and vertical components of motion separately. 2) Derive an equation for horizontal displacement as a function of time. 3) Derive an equation for vertical displacement as a function of time. 4) Solve the horizontal displacement equation for time. 5) Substitute that expression of time into the vertical displacement equation, resulting in vertical displacement as a function of horizontal displacement (trajectory).

Resistance is Futile

Negligible Air Resistance: Horizontal Motion

With no air resistance, there is no force acting in the horizontal direction, so the projectile’s horizontal velocity remains constant for as long as its in flight. From kinematics, horizontal displacement: x = Vx0t = V0 cos (θ)t Initial velocity vector and its components.

Resistance is Futile

Negligible Air Resistance: Vertical Motion

In the vertical direction, the acceleration of gravity will cause the projectile’s vertical velocity to decrease throughout its flight. From kinematics, vertical displacement: y = Vy0t − 1 2gt2 = V0 sin (θ)t − 1 2gt2 Initial velocity vector and its components.

Resistance is Futile

Negligible Air Resistance: Trajectory

Horizontal displacement: x = Vx0t Vertical displacement: y = Vy0t − 1 2gt2

Resistance is Futile

Negligible Air Resistance: Trajectory

Horizontal displacement: x = Vx0t Solve for time (t): t = x Vx0 Vertical displacement: y = Vy0t − 1 2gt2

Resistance is Futile

Negligible Air Resistance: Trajectory

Horizontal displacement: x = Vx0t Solve for time (t): t = x Vx0 Vertical displacement: y = Vy0t − 1 2gt2 Substitute into vertical displacement: Yvac = Vy0 x Vx0

• − g

2 x Vx0 2

Resistance is Futile

Negligible Air Resistance: Trajectory

Horizontal displacement: x = Vx0t Solve for time (t): t = x Vx0 Vertical displacement: y = Vy0t − 1 2gt2 Substitute into vertical displacement: Yvac = Vy0 x Vx0

• − g

2 x Vx0 2 Simplify to reveal: Yvac = x tan θ − gx2 2V 2

0 cos2 θ

Resistance is Futile

Trajectory With Negligible Air Resistance

V0 = 60m/s, θ = 45◦

Resistance is Futile

Strategy for Deriving Trajectory

The same strategy will be used to derive trajectory through air: 1) Analyze the horizontal and vertical components of motion separately. 2) Derive an equation for horizontal displacement as a function of time. 3) Derive an equation for vertical displacement as a function of time. 4) Solve the horizontal displacement equation for time. 5) Substitute that expression of time into the vertical displacement equation, resulting in vertical displacement as a function of horizontal displacement (trajectory).

Resistance is Futile

Linear Air Resistance: Horizontal Net Force

Begin with an analysis of the forces acting in the horizontal direction, where the only force is the force of air resistance. Begin with the Newtonian equation:

• Fx = max

max = −bVx mdVx dt = −bVx dVx dt = −bVx m (1) Equation (1) is the differential equation to be solved for Vx. Free body diagram

• f projectile in

flight.

Resistance is Futile

Linear Air Resistance: Horizontal Initial Conditions

At t = 0, the velocity in the horizontal direction, Vx0, will be equal to the initial velocity times the cosine of the initial projection angle (V0 cos θ). These initial conditions complete the setup of the differential equation for horizontal velocity. dVx dt = −bVx m , Vx(0) = Vx0

Resistance is Futile

Linear Air Resistance: Horizontal Velocity

Solving by separation of variables shows: Vx(t) = Vx0e−bt/m = V0 cos(θ)e−bt/m Horizontal velocity, where V0 = 60m/s, θ = 45◦, m = 1kg, b = 1kg/s.

Resistance is Futile

Linear Air Resistance: Horizontal Displacement

Since velocity is the time derivative of displacement, integrating the equation for horizontal velocity yields horizontal position: x(t) = t Vx0e−bt′/mdt′ = mVx0 b (1 − e−bt/m) Horizontal position, where V0 = 60m/s, θ = 45◦, m = 1kg, b = 1kg/s.

Resistance is Futile

Linear Air Resistance: Vertical Net Force

To derive the differential equation

• f vertical velocity, refer again to

the free body diagram and set up the Newtonian equation:

• Fy = may

may = −mg − bVy mdVy dt = −mg − bVy dVy dt = −g − bVy m

Resistance is Futile

Linear Air Resistance: Vertical Initial Conditions

When t = 0, the projectile is moving in the vertical direction with a velocity equal to the initial velocity times the sine of the projected angle (V0 sin θ). For ease, Vy0 will be used to express this. dVy dt = −g−bVy m , Vy(0) = Vy0

Resistance is Futile

Linear Air Resistance: Vertical Velocity

Solving, again by separation of variables, shows: Vy(t) = mg b + Vy0

• e−bt/m − mg

b Vertical velocity, where V0 = 60m/s, θ = 45◦, m = 1kg, b = 1kg/s.

Resistance is Futile

Linear Air Resistance: Vertical Displacement

Attain an equation for vertical displacement by integrating the equation of vertical velocity: y(t) = t

• (mg

b + Vy0)e−bt′/m − mg b

• dt′

= m2g + bmVy0 b2

• (1 − e−bt/m) − mg

b t Vertical position, where V0 = 60m/s, θ = 45◦, m = 1kg, b = 1kg/s.

Resistance is Futile

Linear Air Resistance: Trajectory

Horizontal Displacement: x = mVx0 b (1 − e−bt/m), Vertical Displacement: y = m2g + bmVy0 b2

• (1 − e−bt/m) − mg

b t

Resistance is Futile

Linear Air Resistance: Trajectory

x = mVx0 b (1 − e−bt/m), y = m2g + bmVy0 b2

• (1 − e−bt/m) − mg

b t Solve horizontal displacement for time (t): t = m b ln

• mVx0

mVx0 − bx

Resistance is Futile

Linear Air Resistance: Trajectory

x = mVx0 b (1 − e−bt/m), y = m2g + bmVy0 b2

• (1 − e−bt/m) − mg

b t Solve horizontal position for time (t): t = m b ln

• mVx0

mVx0 − bx

• Substitute this expression of t into the equation for vertical

position to produce an equation for trajectory (Y ). Y = mg bVx0 + Vy0 Vx0

• x + m2g

b2 ln

• 1 −

b mVx0 x

Resistance is Futile

Linear Air Resistance: Trajectory

Expand Vx0, and Vy0: Y = mg sec θ bV0 + tan θ

• x + m2g

b2 ln

• 1 − b sec θ

mV0 x

Resistance is Futile

What is b?

Recall the differential equation for the vertical component of velocity: dVy dt = −g − bVy m

Resistance is Futile

What is b?

dVy dt = −g − bVy m An object dropped from rest and allowed to fall freely will accelerate until reaching its terminal velocity, at which point acceleration equals zero. Under these conditions, and letting the y-axis point downward, we can write: 0 = g − bVterm m

Resistance is Futile

What is b?

Solving for b shows: b = mg Vterm

Resistance is Futile

What is b?

Now, consider a 3.2 gram paintball with an experimentally derived terminal velocity of 21.8 meters per second: b = mg Vterm = (0.0032kg)(9.81m/s2) 21.8m/s = 0.00144kg/s

Resistance is Futile

Trajectory of a Paintball

Recall the equation for trajectory: Y = mg sec θ bV0 + tan θ

• x + m2g

b2 ln

• 1 − b sec θ

mV0 x

Resistance is Futile

Trajectory of a Paintball

With values: V0 = 90m/s, θ = 45◦, m = 0.0032kg, b = 0.00144kg/s, g = 9.81m/s2 The equation for a paintball’s trajectory becomes: Y = 1.343x+48.44 ln(1−0.007071x)

Resistance is Futile

Trajectory of a 3.2 gram paintball fired at 90m/s at 45 degrees.

Resistance is Futile

Trajectory of a Paintball With and Without Air Resistance

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Maximum Range Angles

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Increase Mass: 3.2g

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Increase Mass: 10g

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Increase Mass: 15g

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Increase Mass: 25g

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Increase Mass: 50g

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Increase Mass: 100g

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Increase Mass: 10kg

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b Changes with Mass

But a higher mass doesn’t imply it’s OK to assume air resistance is negligible, because b changes with mass. Consider a 7.62kg bowling ball with a terminal velocity of 83.1m/s: b = mg Vterm = (7.62kg)(9.81m/s2) 83.1m/s = 0.8995kg/s

Resistance is Futile

Bowling Ball Trajectory

Resistance is Futile

Bibliography

(1) Pruneau, Claude A. ”Projectile Motion.” Wayne State University. http://rhig.physics.wayne.edu/ pruneau/Courses/PHY5200/lectures/ PHY5200-Chap2.pdf