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Terminal Velocity
Activity Information
Learning Goals
- $\Sigma F = 0$ does not mean no motion
- Relationship between terminal velocity and net force
- Use a computer simulation to model the impact of proposed solutions to a complex real-world problem with numerous criteria and constraints on interactions within and between systems relevant to the problem HS-ETS1-4
- Analyze data to support the claim that Newton's second law of motion describes the mathematical relationship among the net force on a macroscopic object, its mass, and its accelerations HS-PS2-1
- Assessment Boundary: limited to one-dimensional motion and to macroscopic objects moving at non-relativistic speeds
Prior Knowledge Required
- Newton's second law of motion
- Drag equation
- $D=\dfrac{1}{2}C\rho v^2A$
- Terminal velocity
- $v=\sqrt{\dfrac{2W}{C\rho A}}$
- Gravitational force
Code Manipulation
- Copying/pasting code
- Creating code from scratch
- Translating equations into code
- While loops and if statements
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Activity
Handout
Terminal Velocity A coin is being dropped from the top of a building. When will it reach terminal velocity? How does the force of gravity compare to the force of air resistance? Take a look at the unfinished code and answer the following questions:
Pre-coding Questions
- Create a graph showing how you think the velocity of the coin will change over time
- Create a graph showing how you think the forces of gravity, air resistance, and net force change as the coin approaches the ground.
- Using information provided by the code, calculate the terminal velocity of the falling coin. Label each part of the terminal equation, including units.
Post-coding Questions
- Describe your contributions to the coding process within your group.
- What did you find the terminal velocity to be based on the graph produced by the code? Does this agree with your calculations?
- Reflect on your predicted graphs. Explain how they were similar/different than your coding graphs. Revise your graphs as needed.
Code
GlowScript 2.8 VPython #Window Setup scene.width = 500 scene.height = 400 #Objects ground = box(pos=vec(0,-250,0), size=vec(700,100,2), texture="https://images.pexels.com/photos/207204/pexels-photo-207204.jpeg?auto=compress&cs=tinysrgb&h=750&w=1260" ) coin = cylinder(pos=vector(0,0,0), axis=vector(0.5,0,0), radius=15.305, textures=textures.metal) coin.rotate(angle=90, axis=vector(1,1,1)) #Initial Object Properties and Constants (time and time step) g = vec(0,-9.81,0) #Acceleration due to Gravity coin_m = 0.01135 #mass of the object drag_co = 1.15 #Drag Coefficient of t = 0 #Initial Time dt = 0.01 #Time Step tf = 0 #Final Time air_density = 1.2754 #Density of the air coin_r = .015305 Fgrav = vec(0,0,0) Fdrag = vec(0,0,0) #Initial Velocity (coin_v) and Momentum Vectors (coin_p) #coin_v = #coin_p = #Create an arrow to visualize the particle's forces #Set up the graphs force_comparison_graph = graph(title='Comparison of Forces on a Coin Reaching Terminal Velocity',xtitle='time (s)',ytitle='Force (N)',fast=False,width=1000) velocity_graph = graph(title='Velocity of a Coin in Freefall Over Time',xtitle='time (s)', ytitle='Velocity (m/s)',fast=False,width=1000) #Drag and Fgrav and Fnet on the same plot Fdrag_f = gcurve(graph=force_comparison_graph,color=color.red,width=4,markers=False,label='Drag Force') Fgrav_f = gcurve(graph=force_comparison_graph,color=color.blue,width=4,markers=False,label='Fgrav') Fnet_f = gcurve(graph=force_comparison_graph,color=color.purple,width=4,markers=False,label='Fnet') #Velocity on its own plot velocity_f = gcurve(graph=velocity_graph,color=color.green,width=4,markers=False,label='Coin Velocity') #Calculation Loop while True: #Change this so the coin stops when it hits the ground rate(100) #Make the coin move #Keep the arrows representing the particle's velocity and acceleration #Plot on the graphs velocity_f.plot(t,coin_v.y) Fdrag_f.plot(t,Fdrag.y) Fgrav_f.plot(t,Fgrav.y) Fnet_f.plot(t,Fnet.y) t += dt #Advance the time by 1 time step
Answer Key
Handout
Pre-coding Answers
- INSERT PICTURE
- INSERT PICTURE
- The equation for terminal velocity is $V=\sqrt{\dfrac{2W}{C\rho A}}$, where
- $W$ is the weight of the coin (N)
- this is calculated by multiplying the mass of the coin by the acceleration of gravity: $W= 0.01135 \text{kg} * 9.81 \text{m/s/s} = 0.111 \text{N}$
- $C$ is the drag coefficient (1.15, no units)
- $\rho$ is the density of air (1.2754 kg/m³)
- $A$ is the area of the coin
- this is calculated by squaring the radius of the coin and multiplying it by pi: $A= (0.015305 \text{m})^2 * \pi = 0.000736 \text{m²}$
- $v=\sqrt{\dfrac{2*0.111}{1.15*1.275*0.000736}}=14.3 \text{m/s}$