Worksheet for Exploration 2.1: Compare Position vs. Time and Velocity vs. Time Graphs


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1 Worksheet for Exploration 2.1: Compare Position vs. Time and Velocity vs. Time Graphs Shown are three different animations, each with three toy monster trucks moving to the right. Two ways to describe the motion of the trucks are position vs. time graphs and velocity vs. time graphs (position is given in centimeters and time is given in seconds). Restart. Answer the following questions that focus on the velocity and acceleration of the monster trucks. i. Fill out the following tables describing information for each animation. Animation 1 Initial Position Initial Velocity Acceleration Red Green Blue Animation 2 Initial Position Initial Velocity Acceleration Red Green Blue Animation 3 Initial Position Initial Velocity Acceleration Red Green Blue
2 a. How does the initial position affect the various graphs? b. Describe the motion of the trucks by analyzing the position vs. time graphs. i. For each animation write out an equation describing position vs. time for each car. Animation 1 x(t) red = x(t) green = x(t) blue = Animation 2 x(t) red = x(t) green = x(t) blue = Animation 3 x(t) red = x(t) green = x(t) blue =
3 c. Once you have completed (a) and (b), check your answers by analyzing the velocity vs. time graphs. i. For each animation write out an equation describing position vs. time for each car. Animation 1 v(t) red = v(t) green = v(t) blue = Animation 2 v(t) red = v(t) green = v(t) blue = Animation 3 v(t) red = v(t) green = v(t) blue =
4 Worksheet for Exploration 2.2: Determine the Correct Graph a. View the animation of the red ball by selecting Ball Only and describe its motion in words (position is given in meters and time is given in seconds). Restart. i. Record the position vs. time information in the table below for the six "shadows". Position Time ii. For each consecutive set of points determine the displacement between shadows (you will have five). Note displacement is not the same as position. Displacement iii. How much time does it take for each displacement to occur?
5 b. Now view the three possible position vs. time graphs A, B, and C by clicking the links in the table. Which one is the correct graph? Give at least one reason why each of the other two graphs is incorrect. A Correct Plot: Yes or No Discuss what is incorrect B C c. Now view the three possible velocity vs. time graphs D, E, and F by clicking the links in the table. Which one is the correct graph? Give at least one reason why each of the other two graphs is incorrect. D Correct Plot: Yes or No Discuss what is incorrect E F
6 Worksheet for Exploration 2.3: A Curtain Blocks Your View of a Golf Ball A putted golf ball (not shown to scale) rolls on a green. A black curtain blocks your view of the ball, but otherwise does not interfere with the ball's motion (position is given in meters and time is given in seconds). The animation is a side view of the ball rolling on the green. Analyze the velocity vs. time graph for the ball and describe the terrain of the green behind the curtain that blocks your view. Give valid reasons for your answer. Restart. i. The graph can be divided into five periods of time. What is the acceleration for each? a 1 = a 2 = a 3 = a 4 = a 5 = ii. Physically what can happen behind the screen to change the acceleration? After you answer this question, check the correct answer.
7 Worksheet for Exploration 2.4: Set the x(t) of a Monster Truck By now you have seen the equation x = x 0 + v 0 *t + 0.5*a*t 2. Perhaps you have even derived it for yourself. But what does it really mean for the motion of objects? Restart. The animation allows you to explore all three terms in the equation: the initial position by changing x 0 from 50 cm to 50 cm, the velocity term by changing v 0 from 15 cm/s to 15 cm/s, and the acceleration term by changing a from 5 cm/s 2 to 5 cm/s 2. Use the animation to guide your answers to the following questions (position is given in centimeters and time is given in seconds). a. How does changing the initial position affect the position vs. time graph? i. Make a sample set of graphs to justify your answer in (a).
8 b. How does changing the initial position affect the velocity vs. time graph? i. Make a sample set of graphs to justify your answer in (b). c. How does changing the initial velocity affect the velocity vs. time graph? i. Make a sample set of graphs to justify your answer in (c).
9 d. How does a positive initial velocity vs. a negative initial velocity affect the velocity vs. time graph? ii. Make a sample set of graphs to justify your answer in (d).
10 Worksheet for Exploration 2.5: Determine x(t) and v(t) of the Lamborghini a. Find the position of the toy Lamborghini as a function of time, x(t), for each animation (position is given in centimeters and time is given in seconds). Restart. Note that the graph depicts the position as a function of time. Use the "check function" button to see the actual position vs. time graph and use this as a guide for your analysis. 1a 2a 3a 4a Initial position Initial Velocity Acceleration b. Find the velocity of the toy Lamborghini as a function of time, v(t), for each animation (position is given in centimeters and time is given in seconds). Use the "check function" button to see the actual velocity vs. time graph and use this as a guide for your analysis. (If you have taken calculus, this exercise should be particularly straightforward.) 1b 2b 3b 4b Initial Velocity Acceleration
11 Worksheet for Exploration 2.6: Toss the Ball to Barely Touch the Ceiling To show your coordination, you try to toss a ball straight upward so that it just barely touches the ceiling (position is given in meters and time is given in seconds). What initial velocity is required? In this Exploration the acceleration of the ball is 9.8 m/s 2. Calculate this initial velocity and then test your answer by typing the initial velocity in the text box and clicking the "set velocity and play" button. Restart. i. Calculate the velocity required to just have the ball reach the ceiling. Check to see that this is correct by running the animation. V launch = ii. For the speed in i, how fast is the ball moving halfway to the ceiling. Explain this result. v half height = iii. At what time do you expect the ball to move at half the speed that it was launched at, and what height is that? Be able to predict these results. t half speed = height =
12 Worksheet for Exploration 2.7: Drop Two Balls; One with a Delayed Drop Two giant tennis balls are released from rest at a certain height. One (the ball on the right) can be dropped after the first ball is dropped. You may change the time delay from 0 to 2.5 s (enter the time delay in the text box and click the "set delay and play" button). The ghost images mark the balls' positions every 0.5 s (position is given in meters and time is given in seconds). Restart. Choose a one second delay (for simplicity) and then answer the following questions. a. Once the second tennis ball (the ball on the right) is released, does the difference in the speeds increase, decrease, or stay the same? i. Sketch a velocity vs. time plot for each object noting on each the time the balls strike the ground. What is the meaning of the area under each plot? b. Once the second tennis ball (the ball on the right) is released, does their separation increase, decrease, or stay the same? c. Is the time interval between the instants at which they hit the ground smaller than, equal to, or larger than the time interval between the instants at which they were released?
13 Worksheet for Exploration 2.8: Determine the Area Under a(t) and v(t) A 1.0kg cart on a track experiences several different constant accelerations as shown in the animation (position is given in meters and time is given in seconds). The red dot shows you where position measurements are taken. In addition, the graph of either the acceleration vs. time or the velocity vs. time is shown (use the check box to toggle between the two) along with data in a table. One cell of the table shows the calculation of the area under the curve (the integral a dt or v dt) as it is plotted in the graph shown. Restart. View all five animations and answer the questions below for the acceleration vs. time graph. a. What is the initial velocity in each animation? b. What is the final velocity in each animation? c. What is the difference between the final velocity and the initial velocity (vv 0 ) in each animation? d. What is the total area under the curve calculated during each animation? e. How are your answers for (c) and (d) related? Does this make sense? Why?
14 View all five animations and answer the questions below for the velocity vs. time graph (use the check box to view the velocity vs. time graphs). f. What is the initial position in each animation? g. What is the final position in each animation? h. What is the displacement of the cart (xx 0 ) in each animation? i. What is the total area under the curve calculated during each animation? j. How are your answers for (f) and (g) related? Does this make sense? Why?
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