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repository:beating_the_train [2020/02/27 21:31] porcaro1 created |
repository:beating_the_train [2021/02/18 19:12] (current) porcaro1 [Answer Key] |
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| ===Prior Knowledge Required=== | ===Prior Knowledge Required=== | ||
| * Units | * Units | ||
| - | * Dimensional Analysis | ||
| * Kinematic Equations | * Kinematic Equations | ||
| * Newton's 2nd Law of Motion | * Newton's 2nd Law of Motion | ||
| Line 14: | Line 13: | ||
| ===Code Manipulation=== | ===Code Manipulation=== | ||
| * Modify existing code | * Modify existing code | ||
| - | * [[https://www.glowscript.org/#/user/kstedman/folder/Public/program/CollidingCars|Colliding Crates]] | + | * [[colliding_crates|Colliding Crates]] |
| ---- | ---- | ||
| ====Activity==== | ====Activity==== | ||
| ===Handout=== | ===Handout=== | ||
| + | {{ :repository:beating_the_train1.png?nolink&600|}} | ||
| + | **Beating the Train**\\ | ||
| + | |||
| + | A train is approaching a crossing, traveling at a constant velocity if 15 m/s to the east. You are on an afternoon ride on your motorcycle when you get stopped at a stoplight 50 m south of the crossing. At the instant the light turns green, the train is 90 m west of the crossing. Your mass, including the motorcycle, is 400 kg | ||
| + | |||
| + | - Determine the amount of time required for the train to reach the crossing. | ||
| + | - Using the time found in #1, determine the minimum acceleration required for you and your motorcycle to beat the train and safely cross to the other side. | ||
| + | - Given a frictional force of 100 N that opposes you as you accelerate to the train crossing, what is the minimum force exerted by the motorcycle that is necessary to miss the train? | ||
| + | - Modify the [[https://www.glowscript.org/#/user/kstedman/folder/Public/program/CollidingCars| Colliding Crates]] program to fit this scenario. | ||
| ===Code=== | ===Code=== | ||
| + | [[https://www.glowscript.org/#/user/porcaro1/folder/RepositoryPrograms/program/BeatingtheTrain-Incomplete| Link]] | ||
| + | <code Python [enable_line_numbers="true", highlight_lines_extra="0"]> | ||
| + | GlowScript 2.7 VPython | ||
| + | #Creating the objects | ||
| + | floor = box(pos=vector(0,-30,0), size=vector(100,4,12), color=color.white) #I've created the floor that the crate will slide across | ||
| + | crate = box(pos=vector(0,0,0), size=vector(20,20,5), color=color.red) #I've created the crate, along with its dimensions and initial position resting on the floor | ||
| + | |||
| + | #Setting the time interval | ||
| + | t=0 #I've set the initial time to zero. | ||
| + | tf=0.940 #I've set the final time to 0, which gives the crate enough time to slide across the floor | ||
| + | dt=0.01 #I want my time interval set at 1/100th of a time unit | ||
| + | |||
| + | #Creates velocity vectors as a function of time | ||
| + | get_library('https://rawgit.com/perlatmsu/physutil/master/js/physutil.js') #The program needed to know what a motion map is defined as | ||
| + | motionMap = MotionMap(crate, tf, 5, markerScale=0.1) #I want to display 5 arrows showing the motion of the crate | ||
| + | |||
| + | #Giving the objects an initial velocity | ||
| + | cratev=vector(75,0,0) #I'm defining the constant velocity of my crate to be 75 in the x-direction(left to right) | ||
| + | while crate.pos.x<35: #I want the crate to stop before it slides off the floor | ||
| + | rate(50) #This rate can speed up or slow down the replay | ||
| + | crate.pos=crate.pos+cratev*dt #I'm moving the crate by adding the change in position (cratev*dt) to the previous position (crate.pos) | ||
| + | t=t+dt #I'm updating the time | ||
| + | | ||
| + | #This updates the velocity vectors | ||
| + | motionMap.update(t,cratev) #This updates the motion map and display of the arrows as the crate slides across the floor | ||
| + | |||
| + | #This creates the graph of the kinetic energy of the crate | ||
| + | #f1 = gcurve(color=color.blue) #Setting up a graph to show the kinetic energy of the crate as a function of time | ||
| + | #for t in arange(0, 0.94, 0.01): # Time goes from 0 to 0.94 in 0.01 time intervals | ||
| + | # f1.plot(pos=(t,cratev.mag**2)) | ||
| + | </code> | ||
| ---- | ---- | ||
| ====Answer Key==== | ====Answer Key==== | ||
| ===Handout=== | ===Handout=== | ||
| + | {{ :repository:beating_the_train2.png?nolink&600|}} | ||
| + | - To find the time it will take for the train to reach the crossing, divide the distance between the train and the crossing by the velocity of the train: | ||
| + | - $t=\dfrac{90\text{m}}{15\text{m/s}}=6\text{s}$ | ||
| + | - We can use the following kinematic equation to solve for the minimum acceleration required to beat the train: $x=x_{0}+v_{0}t+\dfrac{1}{2}at^2$ | ||
| + | - where: | ||
| + | - $x$ is your final position | ||
| + | - You must travel 50 m | ||
| + | - $x_{0}$ is your initial position | ||
| + | - We can assume we start at 0 m | ||
| + | - $v_{0}$ is your initial velocity | ||
| + | - you start from rest, so $v_{0}=0$ m/s | ||
| + | - $a$ is your acceleration | ||
| + | - We will solve for this | ||
| + | - $t$ is the time you have to beat the train | ||
| + | - We just calculated this to be 6 seconds | ||
| + | - Rearranging to solve for acceleration: | ||
| + | - $a=\dfrac{2(x-x_{0}-v_{0}t)}{t^2}$ | ||
| + | - Plugging in our known values: | ||
| + | - $a=\dfrac{2(50\text{m}-0\text{m}-0\text{m/s}*6\text{s})}{(6\text{s})^2}=\dfrac{100\text{m}}{36\text{s}^2}=2.78\text{m/s²}$ | ||
| + | - To find out the minimum force your motorcycle must exert, we must first create a free body diagram. From this diagram, we know that the sum of the frictional force $F_{f}$ and the force of your motorcycle $F_{m}$ must be equal to the total mass of you and the motorcycle (400 kg) multiplied by the acceleration we just calculated (2.78 m/s²). Mathematically, this looks like: | ||
| + | - $F_{f}+F_{m}=ma$ | ||
| + | - Solving for the force of your motorcycle: | ||
| + | - $F_{m}=ma-F_{f}$ | ||
| + | - Plugging in our known values: | ||
| + | - $F_{m}=400\text{kg}*2.78\text{m/s²}-(-100\text{N})=1212\text{N}$ | ||
| + | - Note that $F_{f}$ is negative because it points in the negative y-direction. This is determined from your free body diagram | ||
| + | - See below | ||
| ===Code=== | ===Code=== | ||
| + | [[https://www.glowscript.org/#/user/porcaro1/folder/RepositoryPrograms/program/BeatingtheTrain-Solution | Link]] | ||
| + | <code Python [enable_line_numbers="true", highlight_lines_extra="3,4,16,17,18,19,20,21,22,23"]> | ||
| + | GlowScript 2.7 VPython | ||
| + | #Creating the objects | ||
| + | train = box(pos=vector(-100,-5,0), size=vector(20,5,8), color=color.red) #I've created the crate, along with its dimensions and initial position resting on the floor | ||
| + | cycle = box(pos=vector(0,-51,0), size=vector(2,2,2), color=color.white) | ||
| + | |||
| + | #Setting the time interval | ||
| + | t=0 #I've set the initial time to zero. | ||
| + | dt=0.01 #I want my time interval set at 1/100th of a time unit | ||
| + | |||
| + | #Creates velocity vectors as a function of time | ||
| + | get_library('https://rawgit.com/perlatmsu/physutil/master/js/physutil.js') #The program needed to know what a motion map is defined as | ||
| + | #motionMap1 = MotionMap(crate1, tf, 5, markerScale=0.1) #I want to display 5 arrows showing the motion of the crate | ||
| + | #motionMap2 = MotionMap(crate2, tf, 5, markerScale=0.1) | ||
| + | #Giving the objects an initial velocity | ||
| + | trainv=vector(15,0,0) #I'm defining the constant velocity of my crate to be 75 in the x-direction(left to right) | ||
| + | cyclev=vector(0,0,0) | ||
| + | cycleacc=vec(0,2.78,0) | ||
| + | while cycle.pos.y<=10: #I want the crate to stop before it slides off the floor | ||
| + | rate(100) #This rate can speed up or slow down the replay | ||
| + | train.pos=train.pos+trainv*dt #I'm moving the crate by adding the change in position (cratev*dt) to the previous position (crate.pos) | ||
| + | cyclev=cyclev + cycleacc*dt | ||
| + | cycle.pos=cycle.pos+cyclev*dt | ||
| + | t=t+dt #I'm updating the time | ||
| + | | ||
| + | #This updates the velocity vectors | ||
| + | #motionMap1.update(t,crate1v) #This updates the motion map and display of the arrows as the crate slides across the floor | ||
| + | #motionMap2.update(t,crate2v) | ||
| + | |||
| + | #This creates the graph of the kinetic energy of the crate | ||
| + | #f1 = gcurve(color=color.blue) #Setting up a graph to show the kinetic energy of the crate as a function of time | ||
| + | #for t in arange(0, 0.94, 0.01): # Time goes from 0 to 0.94 in 0.01 time intervals | ||
| + | # f1.plot(pos=(t,crate1v.mag**2)) # plot time vs. kinetic energy of the crate </code> | ||
| ---- | ---- | ||
| ====See Also==== | ====See Also==== | ||
| + | * [[terminal_velocity | Terminal Velocity]] | ||