This quarter, we have talked a lot about motion. We started with constant motion and our Patrol Buggy. We found objects moving at a constant velocity could be modeled by a linear relationship between position and time. We found that the slope, which tells us how position changes with time, is the velocity. This allows us to model an object with constant velocity using the following mathematical equation:

1                                                                                                                                𝑣!"#$%&#% =𝑥! − 𝑥!Δ𝑡

After solving some problems with this model, we found it would not properly model an object that had a changing velocity (like our cart on a ramp). But we were able to find objects with a changing velocity could be modeled by a linear relationship between velocity and time. We found the slope, which tells us how velocity changes with time, is the acceleration. This allows us to model an object with a uniform acceleration using the following mathematical equation:

2                                                                                                                                              𝑎 =𝑣! − 𝑣!Δ𝑡

Because we know the area under a velocity time graph tells us the change in position 𝚫𝐱 , we were able to find the area under a generic velocity time graph, deriving the following equation:

3                                                                                                              Δ𝑥 = 𝑣! ∙ Δt +12𝑎 ∙ Δt!

Recognizing that it might be useful to model these types of motion without using the variable Δt, we used what we know from equations (2) and (3), we were able to create the following equation: 4                                                                                                                              𝑣!! = 𝑣!! + 2𝑎 ∙ Δx

We have developed the tools to model an object’s motion with position-time, velocity-time, and acceleration-time graphs. We can also model motion using the mathematical equations above. We have the tools to create a position time graph and translate between the different representations. Now we can move on to more advanced types of problem solving! Answer the questions that follow in this packet. When answering these questions focus on which tools you are using to model the motion and find the missing information. We want to try to show as much work as possible and keep our solutions organized. If you decide to use any type of mathematical solution in your calculator, you MUST first list the values of the variables given and write the equation you plan to use (with variables and no numbers)

Name _____________________________________________________ Period ___________________

Acceleration and Two-Dimensional Motion

Posi

tion

Time

Velo

city

Time

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Practice Problems – Using Equations Answer the following questions using the models we have discussed this quarter. If you decide to use any type of mathematical solution in your calculator, you MUST first list the values of the variables given and write the equation you plan to use (with variables and no numbers) 1. A stationary airplane accelerates down a runway at 3.20 m/s/s for 32.8 s until is finally lifts off

the ground. Determine the distance traveled before takeoff.

2. A car starts from rest and accelerates uniformly over a time of 5.21 seconds for a distance of 110 m. Determine the acceleration of the car.

3. Upton Chuck is riding the Giant Drop at Great America. If Upton free falls for 2.6 seconds with an acceleration of 10 m/s/s, what will be his final velocity and how far will he fall?

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5. Gail is driving along at 11.4 m/s when she passes a sign, seeing that she is below the speed limit. She accelerates at 3 m/s/s for 5.8 seconds. What is her final speed, and how far does she travel?

6. A car traveling at 22.4 m/s skids to a stop in 2.55 s. Determine the skidding distance of the car.

7. An engineer is designing the runway for an airport. Of the planes that will use the airport, the

lowest acceleration rate is likely to be 3 m/s2. The takeoff speed for this plane will be 65 m/s. Assuming this minimum acceleration, what is the minimum allowed length for the runway?

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8. A parked dragster accelerates to a speed of 112 m/s in only 7 seconds. How far does the

dragster go in this time?

9. A bullet leaves a rifle with a muzzle velocity of 521 m/s. While accelerating through the barrel of the rifle, the bullet moves a distance of 0.840 m in 0.003 seconds. Determine the acceleration of the bullet.

10. A feather is dropped on the moon from a height of 1.80 meters. The acceleration of gravity on the moon is 1.67 m/s2. Determine the time for the feather to fall to the surface of the moon.

Did you find those problems relatively easy? Then skip to the next page. Need more practice? Work on the following problems:  

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More Advanced Problems

When you solve more advanced problems, you should think carefully about the method(s) you choose to use to solve the problem. You may find that, at times, using equations is an easier way to solve them, while at other times a graph makes the solution easier.

It is HIGHLY recommended that even if you do not draw a fully-to-scale graph, you at least sketch what the x-t and v-t graphs would look like.

11. A driver is speeding along at 30 m/s when he suddenly sees an accident up ahead. His reaction

time is 1.5 seconds (which means that this is how long it takes him to hit the brakes). Once he hits the brakes, he accelerates at a rate of -5 m/s/s. How far does he travel before he comes to a stop?

12. Jon and Fred are in a race. Jon runs at a speed of 1.4 m/s, while Fred runs at a speed of 1.8 m/s. How long does it take before they are 20 meters apart?

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13. Tom and Dan are in a race. Both runners start from rest. Tom accelerates at 0.2 m/s/s, while Dan accelerates at 0.4 m/s/s. How long does it take before they are 20 meters apart?

14. A Volkswagon beetle is driving past a sportscar is stopped. The Volkswagon is moving at a constant velocity of 20 m/s. Just as it passes, the sportscar begins to accelerate at 10 m/s/s. How long does it take the sportscar to catch up?

15. A police officer is driving at a constant speed when he turns on his lights. A car 100 m up ahead pulls off to the side to let the police car pass. The car ahead is traveling at 35 m/s, and begins to slow down immediately, coming to a stop in 11.7 seconds. Just as the car stops, the police car passes. How fast was the police car traveling?

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16. In the 2008 Olympics, Jamaican sprinter Usain Bolt shocked the world as he ran the 100-meter dash in 9.69 seconds. Assume that he accelerated for the first four seconds, and then ran the rest of the race at a constant velocity. How far did he run in the first four seconds, and what was his final speed? (Note: this problem is very difficult. It will take patience and perseverance to solve – see if you can!!!)

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Physics Lab Challenge – Projectiles Your goal is to learn about projectile motion from this packet and then complete the lab challenge below. Lab Challenge: A ball is going to be fired from a mini-cannon, and you need to predict where it will land. You will receive extra credit if you successfully complete the challenge!!! Rules: You may take as many measurements as you want beforehand without changing the angle. Then, you will be assigned a random angle and must place a cup such that the ball will land in it, without taking any additional measurements. You only get one try.

YOU MUST COMPLETE THIS PACKET BEFORE CONDUCTING THE LAB CHALLENGE! Notice the difference between this motion and motion that we have studied before. Previously, we studied one dimensional motion – just back and forth. Now, our object is moving in two dimensions: not just back and forth, but also up and down at the same time. To understand the motion of the metal ball, we must first learn about projectiles – objects that are moving and only being affected by the force of gravity. The next activities are designed to help you think about this kind of motion. Lets not forget the equations we have used to model motion this far:

𝑣!"#$%&#% =𝑥! − 𝑥!Δ𝑡

𝑎 =𝑣! − 𝑣!Δ𝑡

Δ𝑥 = 𝑣! ∙ Δt +12𝑎 ∙ Δt!

𝑣!! = 𝑣!! + 2𝑎 ∙ Δx

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Activity 1 - Try this!

1. First, make a prediction – which penny will hit the ground first? Why? _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 2. Put two pennies on the edge of a table as shown above. Pull back a ruler and then release it, so that it hits both pennies at the same time, but one gets hit harder. One penny will have a greater velocity as it leaves the table compared to the other one. 3. Listen carefully! Which one hit the ground first? _________________________________________ 4. What does this tell you about the vertical motion of the two coins? _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

CHECK THE WRITTEN EXPLANATION BEFORE MOVING ON Ask your teacher if you are not sure where to find this

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Activity 2 – Video Analysis of Projectile Motion Ask your teacher where to find the video. If you have forgotten how to analyze a video in LoggerPro, you can refer to the instructions on my website under Unit 3, Day 2- Video Analysis and Board Meeting” to find the written and video instructions. 1. Plot points on the video, tracking the object as it moves across the video. So far this year, all of our velocities have been in the horizontal direction. But what if something was given velocity that was at an angle? That object would have to be moving in the horizontal direction AND the vertical direction. In this activity, keep in mind that the direction of the velocity matters. We can describe this motion using the horizontal velocity (horizontal component) AND the vertical velocity (vertical component). 2. When you are done plotting points on the video, create a graph of just your horizontal (“X”) position vs. time. You can do this by choosing “X” on the y-axis of your graph. Sketch your graph in the box to the right: 3. Now, create a graph of your horizontal velocity vs. time. You can do this by choosing “X Velocity” on the y-axis 4. What kind of horizontal motion does the projectile have? How do you know?

Sketch X vs. time graph:  

Sketch x-velocity vs. time graph:  

Vx

Vy

Vnet

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5. Now, create a graph of just your vertical (“Y”) position vs. time. You can do this by choosing “Y” on the y-axis of your graph. Sketch your graph in the box to the right: 6. Now, create a graph of your horizontal velocity vs. time. You can do this by choosing “Y Velocity” on the y-axis 7. What is the maximum height that the projectile reaches? How do you know? 8. Label the Y position graph above. Describe the motion in the first part. What about the second part? 9. Label the y-velocity graph above. Describe the motion in the first part. What about the second part? 10. What kind of vertical motion does the projectile have? How do you know? CHECK WITH YOUR TEACHER BEFORE MOVING ON

Don’t skip this step.

Sketch Y vs. time graph:  

Sketch y-velocity vs. time graph:  

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Activity 3: Practice Problems

#1. Imagine that a cannonball is dropped from a height of 20 meters. It has no initial velocity. It accelerates downward with an acceleration of a=-9.8 m/s/s. How long does it take to reach the ground? Show your work. #2. Imagine that the cannonball is now shot horizontally out of a cannon from a height of 20 meters. It has an initial velocity of 18 m/s. a. How long does it take to reach the ground? (If you are stuck on this question, refer back to activity 1) b. How far does it fly horizontally?

CHECK THE ANSWER KEY BEFORE MOVING ON Ask your teacher if you are not sure where to find this

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Activity 4: Simulation You are going to use a simulation to conduct an investigation about projectile motion. Remember that velocity can be broken up into its horizontal and vertical components. In this activity, keep in mind your understanding of the horizontal and vertical components of velocity Open the simulation at http://bit.ly/xovGi9 In this activity, you will be required to do some thinking, planning, and analyzing on your own. You are not always told exactly what to do. Be sure to read and use all the information you have, and think carefully about what you plan to do to test each research question. Question: How does the angle of launch affect the distance the projectile travels? 1. Your prediction: ____________________________________________________________________ 2. How will you test this? Think carefully about your independent, dependent, and control variables.

RUN YOUR PLAN BY YOUR TEACHER BEFORE MOVING ON

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3. Record your data below. You will need to label your table according to what data you plan to collect.

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Question: How does the angle of launch affect the distance the projectile travels? 4. What conclusions can you draw from your investigation? 5. Explain your results in terms of the vertical and horizontal components of velocity:

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Activity 5: Read and Practice A. Go to each of the following websites. Read the first part, then try the problems on your own http://bit.ly/Wx33xj B. Now, go to the following website: http://bit.ly/Wx3mZ0 Answer these questions from the website. Be sure the try the problems on your own first – then check your answer. A ball was kicked at a speed of 10 m/s at 37 degrees from the horizontal and returns to ground level further down the field. #1. What were the horizontal and vertical components of the ball’s velocity? Show your work. #2. How much time did it spend in the air? #3. How far down the field did it land? #4. How high did it rise in the air?

Continued on next page…

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While standing on a 30 meter bridge, a fisherman tosses some unused bait off the bridge at a speed of 10 m/s at an angle of 37 degrees. #5. Notice that the initial velocity is the same as the previous question. What are the horizontal and vertical components? #6. How much time did it spend in the air (this will not necessarily be the same as the previous question!)? #7. How far downrange does it land in the water from the base of the bridge? #8. At what speed did it enter the water?

CHECK  YOUR  ANSWERS  AGAINST  THE  ANSWER  KEY.  CONGRATULATIONS!    YOU  ARE  NOW  READY  FOR  THE  LAB  CHALLENGE!  

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6: More Practice Problems - optional Note: problems from Giancoli’s “Physics Principles with Applications” #1. A diver running 1.8 m/s dives out horizontally from the edge of a vertical cliff and reaches the water below 2.0 sec later. How high was the cliff and how far from its base did the diver hit the water? #2. A football is kicked at ground level with a speed of 17.0 m/s at an angle of 30 degrees to the horizontal. What is its hang time – e.g., how much later does it hit the ground? #3. A shotputter throws the shot with an initial speed of 14 m/s at a 40 degree angle to the horizontal. Calculate the horizontal distance traveled by the shot if it leaves the athlete’s hand at a height of 2.2 meters above the ground. #4. When Babe Ruth hit a homer over the 12 meter left field fence 100 meters from home plate, roughly what was the minimum speed of the ball when it left the bat? Assume the ball was hit 1.0 meters above the ground and its path initially made a 35 degree angle with the ground.