 Energy of a Tossed Ball

When a juggler tosses a bean ball straight upward, the ball slows down until it reaches the top of its path and then speeds up on its way back down. In terms of energy, when the ball is released it has kinetic energy, KE. As it rises during its free-fall phase it slows down, loses kinetic energy, and gains gravitational potential energy, PE. As it starts down, still in free fall, the stored gravitational potential energy is converted back into kinetic energy as the object falls.

If there is no work done by frictional forces, the total energy will remain constant. In this experiment, we will see if this works out for the toss of a ball. In this experiment, we will study these energy changes using a Motion Detector.

objectives

·    Measure the change in the kinetic and potential energies as a ball moves in free fall.

·    See how the total energy of the ball changes during free fall.

Materials

 LabPro or CBL 2 interface   TI Graphing Calculator volleyball, basketball, or other similar fairly heavy ball DataMate program wire basket Vernier Motion Detector

Preliminary questions

For each question, consider the free-fall portion of the motion of a ball tossed straight upward, starting just as the ball is released to just before it is caught. Assume that there is very little air resistance.

1.      What form or forms of energy does the ball have while momentarily at rest at the top of the path?

2.      What form or forms of energy does the ball have while in motion near the bottom of its path?

3.      Sketch a graph of velocity vs. time for the ball.

4.      Sketch a graph of kinetic energy vs. time for the ball.

5.      Sketch a graph of potential energy vs. time for the ball.

6.      If there are no frictional forces acting on the ball, how is the change in the ball’s potential energy related to the change in kinetic energy?

Procedure

1.      Measure and record the mass of the ball you plan to use in this experiment.

2.      Connect the Vernier Motion Detector to the DIG/SONIC 1 port of the LabPro. Use the black link cable to connect the interface to the TI 83+. Firmly press in the cable ends.

3.      Place the Motion Detector on the floor and protect it by placing a wire basket over it.

4.      Turn on the calculator and start the DATAMATE program. Press clear to reset the program.

5.   In this step, you will toss the ball straight upward above the Motion Detector and let it fall back toward the Motion Detector. This step may require some practice. Hold the ball directly above and about 0.5 m from the Motion Detector. Use two hands. Be sure to pull your hands away from the ball after it starts moving so they are not picked up by the Motion Detector. Select START to begin data collection. You will notice a clicking sound from the Motion Detector. Wait one second, then toss the ball straight upward. Be sure to move your hands out of the way after you release it. A toss of 0.5 to 1.0 m above the Motion Detector works well. You will get best results if you catch and hold the ball when it is about 0.5 m above the Motion Detector.

6.   After data collection is complete press enter to view the distance graph.

7.   Examine the distance vs. time graph. Repeat Step 5 if your distance vs. time graph does not show a region of smoothly changing distance. Check with your instructor if you are not sure whether you need to repeat the data collection. To repeat data collection, press enter to return to the SELECT GRAPH menu; select MAIN SCREEN and then START to repeat data collection.

Data Table

 Mass of the ball (kg)

 Position Time Height Velocity PE KE TE (s) (m) (m/s) (J) (J) (J) After release Between release and top Top of path Between top and catch Before catch

Analysis

1.      To analyze only the free-fall portion of the data you must discard all other data:

b.    It is easiest to select the proper region using the velocity graph. Use the up and dpwn cursor keys to select VELOCITY.

c.    Select SELECT REGION.

d.    Using the cursor keys move the flashing cursor to the beginning of the straight-line, negative-slope region. This portion corresponds to free fall.

e.    Press enter to select the left (lower) bound.

f.      Use the cursor keys left and right to move the flashing cursor to the end of the straight-line section, corresponding to free fall.

g.    Press enter to select the right (upper) bound.

h.    Press enter again to see your selected data.

2.   To explore the energy of the ball (kinetic and potential) at various moments during the motion, you can trace across the graph using the cursor keys and read positions and velocities at certain times. Record the following times and velocities in your Data Table: Trace to the moment just after the ball was released into free fall, then when the ball was at the top of the path (where velocity was about zero), and then just before the free-fall motion ended. Choose two more points approximately halfway in time between the three recorded so far. You should now have five times and velocities.

3.   Now read the position of the ball at the same five times. Press enter and choose DISTANCE to see the distance graph. Trace across the graph to the same five times you used in the previous step and record the corresponding distances in the Data Table.

4.   For each of the five points in the Data Table, calculate the Potential Energy (PE), Kinetic Energy (KE), and Total Energy (TE). Use the position of the Motion Detector as the zero of your gravitational potential energy.

5.   How well does this table you have just completed show conservation of energy? Explain.

6.   You can also graph the ball’s energy at all the measured times, instead of just the five in the Data Table. Using the distance data in L6 and mgh, you can store gravitational potential energy at each moment in L3. Using the velocity data in L7, you can use ½mv2 to store the ball’s kinetic energy at each moment in L2. Finally, you can find the total mechanical energy at each moment by finding the sum of the potential and kinetic energies in L2 and L3, storing the result in L4. To do these calculations will require manipulating the data outside of the DATAMATE program, so press enter, MAIN SCREEN, and then QUIT to leave DATAMATE. To calculate the kinetic energy KE using ½mv2, storing the data in L2:

·    The velocity data are stored in list L7. Where m is the mass of your ball in kg, press
0.5 x m x L7^2 sto L2. (On the TI-83 and 83 Plus, to enter L7 you must press 2nd [LIST], and then choose L7 from the list. It will be entered as LL7. To enter L2 or any of the first six lists, press 2nd and then the corresponding digit.

7.   In a similar way you can calculate the ball’s gravitational potential energy (PE) using mgh, where the height h comes from your distance data, storing the result at each time in L3:

·    The distance data are stored in list L6. Where m is the mass of your ball in kg, press 9.8 x  m  x L6  sto L3.

8.   Finally, calculate the ball’s total energy KE + PE and store the result in L4:

·    Press L2 +L3 sto L4.

9.   Plot the KE, PE, and total energy on your calculator:

TI-83 Plus

a.    Press 2nd [stat plot] and select Plot 1. (On the TI-73 press 2nd [PLOT].)

b.    Use the arrow keys to position the cursor on each of the following Plot1 settings. Press Enter to select any of the settings you change: Plot1 = On, Type =graph button, Xlist = L 1, Ylist = L2, and Mark = . This is the kinetic energy.

c.    Press 2nd [stat plot] and select Plot 2.

d.    Use the arrow keys to position the cursor on each of the following Plot2 settings. Press  Enter to select any of the settings you change: Plot2 = On, Type =graph button, Xlist = L 1, Ylist = L3, and Mark = +. This is the potential energy.

e.    Press 2nd [stat plot] and select Plot 3.

f.      Use the arrow keys to position the cursor on each of the following Plot3 settings. Press  Enter to select any of the settings you change: Plot3 = On, Type =graph button, Xlist = L 1, Ylist = L4, and Mark =. This is the total energy.

g.    Press  Zoom and then select ZoomStat (use cursor keys to scroll to ZoomStat) to draw a graph with the x and y ranges set to fill the screen with data.

10.   Inspect your kinetic energy vs. time graph (marked with ) for the free-fall flight of the ball. Explain its shape.

11.   Inspect your potential energy vs. time graph (marked with +) for the free-fall flight of the ball. Explain its shape.

12.   Record the three energy graphs by printing or sketching.

13.         Compare your energy graphs predictions (from the Preliminary Questions) to the real data for the ball toss.

14.   What do you conclude from the total energy vs. time graph (marked with and plotted as a line plot) about the total energy of the ball as it moved up and down in free fall? Does the total energy remain constant? Should the total energy remain constant? Why? If it does not, what sources of extra energy are there or where could the missing energy have gone?

Extensions

1.      What would change in this experiment if you used a very light ball, like a beach ball?

2.      What would happen to your experimental results if you entered the wrong mass for the ball in this experiment?