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 F drag = bV As the term implies, the linear model fairly assumes that the force of 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 F drag = 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: Initial velocity x = V x 0 t vector and its = V 0 cos ( θ ) t 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 = V y 0 t − 1 2 gt 2 Initial velocity = V 0 sin ( θ ) t − 1 2 gt 2 vector and its components.

Resistance is Futile Negligible Air Resistance: Trajectory Vertical displacement: Horizontal displacement: y = V y 0 t − 1 2 gt 2 x = V x 0 t

Resistance is Futile Negligible Air Resistance: Trajectory Horizontal displacement: Vertical displacement: x = V x 0 t y = V y 0 t − 1 Solve for time (t): 2 gt 2 x t = V x 0

Resistance is Futile Negligible Air Resistance: Trajectory Vertical displacement: Horizontal displacement: y = V y 0 t − 1 2 gt 2 x = V x 0 t Substitute into vertical Solve for time (t): displacement: x � x � x t = � 2 � − g V x 0 Y vac = V y 0 V x 0 2 V x 0

Resistance is Futile Negligible Air Resistance: Trajectory Vertical displacement: Horizontal displacement: y = V y 0 t − 1 2 gt 2 x = V x 0 t Substitute into vertical Solve for time (t): displacement: x � x � x t = � 2 � − g V x 0 Y vac = V y 0 2 V x 0 V x 0 Simplify to reveal: gx 2 Y vac = x tan θ − 0 cos 2 θ 2 V 2

Resistance is Futile Trajectory With Negligible Air Resistance V 0 = 60 m / 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 Free body diagram direction, where the only force is of projectile in the force of air resistance. Begin flight. with the Newtonian equation: � F x = ma x ma x = − bV x mdV x = − bV x dt dV x = − bV x (1) dt m Equation (1) is the differential equation to be solved for V x .

Resistance is Futile Linear Air Resistance: Horizontal Initial Conditions At t = 0, the velocity in the horizontal direction, V x 0 , will be equal to the initial velocity times the cosine of the initial projection angle ( V 0 cos θ ). These initial conditions complete the setup of the differential equation for horizontal velocity. dV x = − bV x m , V x (0) = V x 0 dt

Resistance is Futile Linear Air Resistance: Horizontal Velocity Solving by separation of variables shows: V x ( t ) = V x 0 e − bt / m = V 0 cos( θ ) e − bt / m Horizontal velocity, where V 0 = 60 m / s , θ = 45 ◦ , m = 1 kg , b = 1 kg / 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: � t V x 0 e − bt ′ / m dt ′ x ( t ) = 0 Horizontal position, where = mV x 0 (1 − e − bt / m ) V 0 = 60 m / s , θ = 45 ◦ , m = 1 kg , b b = 1 kg / s .

Resistance is Futile Linear Air Resistance: Vertical Net Force To derive the differential equation of vertical velocity, refer again to the free body diagram and set up the Newtonian equation: � F y = ma y ma y = − mg − bV y mdVy = − mg − bV y dt dVy = − g − bV y dt 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 ( V 0 sin θ ). For ease, V y 0 will be used to express this. dVy = − g − bV y m , V y (0) = V y 0 dt

Resistance is Futile Linear Air Resistance: Vertical Velocity Solving, again by separation of variables, shows: � mg e − bt / m − mg � V y ( t ) = b + V y 0 b Vertical velocity, where V 0 = 60 m / s , θ = 45 ◦ , m = 1 kg , b = 1 kg / s .

Resistance is Futile Linear Air Resistance: Vertical Displacement Attain an equation for vertical displacement by integrating the equation of vertical velocity: � t ( mg b + V y 0 ) e − bt ′ / m − mg � � dt ′ y ( t ) = b 0 � m 2 g + bmV y 0 � (1 − e − bt / m ) − mg = b t b 2 Vertical position, where V 0 = 60 m / s , θ = 45 ◦ , m = 1 kg , b = 1 kg / s .

Resistance is Futile Linear Air Resistance: Trajectory Vertical Displacement: Horizontal Displacement: � m 2 g + bmV y 0 � (1 − e − bt / m ) − mg x = mV x 0 (1 − e − bt / m ) , y = b t b 2 b

Resistance is Futile Linear Air Resistance: Trajectory � m 2 g + bmV y 0 � (1 − e − bt / m ) − mg x = mV x 0 (1 − e − bt / m ) , y = b t b 2 b Solve horizontal displacement for time ( t ): t = m � mV x 0 � b ln mV x 0 − bx

Resistance is Futile Linear Air Resistance: Trajectory � m 2 g + bmV y 0 � (1 − e − bt / m ) − mg x = mV x 0 (1 − e − bt / m ) , y = b t b 2 b Solve horizontal position for time ( t ): t = m � mV x 0 � b ln mV x 0 − bx Substitute this expression of t into the equation for vertical position to produce an equation for trajectory ( Y ). � mg x + m 2 g + V y 0 � � b � Y = b 2 ln 1 − x bV x 0 V x 0 mV x 0

Resistance is Futile Linear Air Resistance: Trajectory Expand V x 0 , and V y 0 : x + m 2 g � mg sec θ � � 1 − b sec θ � Y = + tan θ b 2 ln x bV 0 mV 0

Resistance is Futile What is b ? Recall the differential equation for the vertical component of velocity: dVy = − g − bV y dt m

Resistance is Futile What is b ? = − g − bV y dVy dt 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 − bV term m

Resistance is Futile What is b ? Solving for b shows: mg b = V term

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: mg b = V term = (0 . 0032 kg )(9 . 81 m / s 2 ) 21 . 8 m / s = 0 . 00144 kg / s