Honors/AP Lab 2.3 Acceleration, Freefall, Symmetry

Name ______School ______Date ______Honors/AP Lab 2.3 – Acceleration, Freefall, Symmetry Purpose To use the ultrasonic motion detector to analyze position, velocity, and acceleration graphs of an accelerating object. To use Frame Flipper tool to determine the acceleration of an object moving vertically with negligible friction. To investigate symmetries in the motion of an object with constant acceleration Virtual Equipment Virtual dynamics track Frame Flipper software tool and video clips of vertical motion Logger Pro (LP) software PENCIL Discussion In our previous lab concerning constant velocity we found that the motion sensor could measure the distance to an object 31 times a second and plot the data as a graph of position vs. time. With Logger Pro we were also able to calculate the rate of change in position, that is, the velocity, and plot velocity vs. time. The following table summarizes our results so far. (We didn’t actually look at areas during the lab. They’ve been included here for completeness.)

Graph Slope Area Position, time (x,t) velocity ------Velocity, time (v,t) Displacement, x Acceleration, time (a,t) jerk, jolt, surge, lurch Change in velocity, v

We’ve also learned that 1. a straight-line position, time (x,t) graph indicates a velocity 2. a straight-line, horizontal position, time (x,t) graph indicates velocity We now turn our attention to acceleration. We’ll look at position, velocity, and acceleration graphs. The table above reminds us that the slope of the position, time graph is the velocity. The slope of the acceleration, time graph and the area under a position, time graph are not of interest to us. 3. In the table fill in the name that we give to the slope of the velocity, time graph. You now have a complete list of the two slopes and two areas that are of interest to us in this course. Procedure Step 1 Position on an incline: Start up the Dynamics Track apparatus. Your low-friction track has a feature that you may not have noticed. If you move your mouse over the track near either end, your pointer changes and the message box informs you that you can tilt the track by dragging up or down. Give it a try. It’s also important to realize that the x-axis rotates with the track.

Set up your track at an angle between -4° and -10°. Track angle = ______°

Honors/AP Lab 2.3 - Freefall 1 February 17, 2012 We want to observe the motion of the cart moving freely up the ramp (to the left) and then back down. Don’t turn on the motion sensor until you’re asked to. Practice throwing the cart up the ramp (left) with just enough initial velocity to make it most of the way to the top end without hitting the top end bumper. It takes some practice. Actually it’s just really hard to do this. There is an easier way. Under the right end of the track is a collection of controls. The Go button is the one you want. It will start the cart at a repeatable initial velocity. You’ll set it using the numeric stepper under the left end of the Go button, and then press the Go button. With your angle set, try different negative initial velocities until the cart goes almost to the top of the track. Initial velocity, Vo = -______cm/s Discuss with your partner what you’d expect for the shape of a position vs. time graph for this motion – starting at the bottom, up to the top, and then back to the bottom of the ramp. Be sure and consider where the graph will start when the cart is at the bottom, where the graph will be when the cart is at the top, and where it will be when the cart returns to the bottom. Also discuss your prediction for the shape of the graph. You should use words like rise, fall, slow, fast, slowing down, steep, speeding up, positive direction and negative direction. Ignore the periods before and after this motion. 4. Summarize your comments below.

5. Sketch your prediction of the shape of a position, time graph of this motion in Figure 5a. Now try it with the detector on. Sweet! Sketch the resulting position, time graph in Figure 5b. Click “copy data to clipboard” and then open Logger Pro and paste your data into the table as you did in the velocity lab. Save your result as “Graph 5a.” Return to the lab apparatus. Many students find one aspect of the graph counter-intuitive. When the cart is at the bottom of the track, the graph is at the top. This is because x = 0 is at the left end of the track, closest to the sensor, and the maximum position is the point farthest from the sensor, which happens to be at the bottom of the track because of the way it’s tilted. It takes some critical thinking to get a clear picture of how the graph describes the cart’s motion. This is what you’re here in the lab for. Don’t miss the opportunity to nail down these concepts.

5a. Prediction: Up, then down ramp 5b. Observation: Up, then down ramp

What can you read about the car’s position from this graph? The graph starts high, corresponding to a large distance from the detector. As the cart moves up the ramp its position decreases, but remains positive. Then as it rolls back down it reverses itself. The graph is also symmetrical – a very useful property to remember. Step 2 Velocity on an incline We now want to consider the graph of velocity vs. time for the cart. What would that look like? As usual, a concise conversation with your lab partner would be a good idea.

Lab 2.3 - Freefall 2 February 17, 2012 6. Keeping in mind that it starts off traveling in a negative direction, describe the cart’s motion in terms of its changing velocity. Be sure to include the direction of the velocity which at a given instant will be negative, positive, or neither. Also include the magnitude, which will be large when the cart is going fast, small when it’s going slow, and zero when it’s at rest.

7. Sketch your prediction of the shape of a velocity, time graph you’d expect in Figure 7a.

Return to Logger Pro and clean up your position vs. time graph – add title, units, etc. Add a velocity, time graph below it. (If you have trouble, review the constant speed lab.) Be sure to label everything properly. You may need to adjust the maximum and minimum values of the velocity axis so that the interval where it was freely moving almost fills the graph vertically. (E.g., -2 m/s to +2 m/s) Sketch your observed graph in Figure 7b. Any surprises? It seems like the graph should change when the cart gets to the top, but instead it just passes through the time axis and continues to rise. Look at your graph and your answer to #6. It’s very important that your words say the same thing as the graph. It’s also important that you convince yourself that the graph makes sense. Feel free to change your answer to #6. 8. Clearly describe this graph. Also comment on what is happening to the cart at the instant when the line crosses the velocity axis. You should know what we’re looking for now.

9. Does the cart have a velocity of zero for an instant at the top of its motion or at an instant at the top of its motion? Explain.

Step 3 Acceleration on an incline: 10. Look back at the table on page 1. How do we find acceleration from a velocity, time graph?

11. Based on that answer you should be able to look at your velocity vs. time graph and predict the shape of an acceleration vs. time graph. Discuss your prediction with your partner, and then sketch your prediction of the shape of the cart’s acceleration vs. time graph in Figure 11a. Now use LP to create the acceleration, time graph. You’ll do this by taking the derivative of the velocity with respect to time. Remember, we found that the term derivative refers the rate of change of some quantity. The derivative of the position with respect to the time is Δx/Δt which is the velocity. The derivative of the velocity with respect to time is Δv/Δt which is the acceleration. As before, you’ll need to resize your first two graphs and insert a third graph to display the acceleration.

Lab 2.3 - Freefall 3 February 17, 2012 Sketch LP’s graph in Figure 10b.

You may have been surprised by what you saw. Shouldn’t the acceleration be different going up vs. going down? This is good stuff here. It has to do with the sign (direction) of the velocity. Let’s make sure you’re clear about what’s happening. Here is a pair of statements that are helpful in understanding the sign of the acceleration:  When an object is speeding up, its velocity and acceleration have the same signs.  When an object is slowing down, its velocity and acceleration have opposite signs. Again, don’t miss the chance to nail down these concepts as you go. This is where you learn them. It would be wise to look at our definition of acceleration (Δv/Δt) to see why the sign of the acceleration works this way. It’s important. 12. When the cart is rolling up the ramp, its velocity is negative since it’s moving in the negative direction. It’s slowing down. The sign of its acceleration is 13. When the cart is rolling down the ramp, its velocity is positive since it’s moving in the positive direction. It’s speeding up. The sign of its acceleration is You now have two graphs that can easily provide values for the acceleration of the cart. The acceleration, time graph has a varying height that clearly varies around a certain average value during the period of constant acceleration. Select this part of the graph by dragging across it. Click the Statistics (STAT) icon to find the average (mean) value of the acceleration. Record it below. (The wiggle in the acceleration graph is due to the position → velocity → acceleration calculations.) Another way of determining the acceleration is from the slope of the line of best fit of the velocity graph. (Remember, drag across the velocity graph where the acceleration (slope) is constant and click the Linear Fit icon.) The slope is the acceleration. Record your results below. Use LP to find each value. 14. Acceleration from acceleration, time graph m/s2 15. Acceleration from velocity, time graph m/s2 Step 4: Acceleration on an incline → acceleration due to gravity, g When Galileo studied acceleration along a ramp he saw that the acceleration increased with the angle of the ramp. He wasn’t able to test his measurements given the limitations of using a water clock to measure time, but he did predict that some maximum acceleration would be reached when the ramp was tilted 90°. We refer to this vertical acceleration of a freely falling object as the acceleration due to gravity, g. Later in the course we'll show that that g should equal a/tan(θ) where a is the acceleration for some angle of the ramp, θ. Let’s see what your results would predict for g. Determine this value using the average of your two acceleration values.

2 16. Predicted maximum acceleration, aavg/tan(θ) = ______m/s

Lab 2.3 - Freefall 4 February 17, 2012 Show calculations here.

Compare this result to the value assumed in the lab, 9.80 m/s2 by finding the percentage error. 17. Percentage error % Show calculations here.

Step 5: Freefall With our more modern equipment we can find the acceleration due to gravity, g, and observe some interesting symmetries found when objects move with constant acceleration. We’ll be working with a video of a billiard ball tossed upward. We’ll use a software tool that we call the Frame Flipper. This coding for this tool was written in 1996 by Joe Pearson. He currently (2009) is Application Architect at The Weather Channel. Cool, I think. A Dropped Object Click the “2.3 – Acceleration, Free-fall Lab” link in the Chapter 2 Labs box to open up the second part of this activity. In the first column you’ll see a QuickTime movie of a dropped billiard ball which falls through a distance of about 2 meters. Click the Play button at the bottom of the movie. You can also double-click in the movie to start it. After clicking in the movie you can also use your left and right keyboard arrows to step through the movie backwards and forwards. 1. Briefly describe what you observe in this movie. Use physics lingo!

An Object Thrown Upward Now play the movie of the object thrown upward. This time you'll see the very smooth motion of a billiard ball thrown upward and then allowed to fall back down. 2. How does the trip upward compare to the trip downward in this movie?

3. Notice how the spacing between the images and the fuzziness of the images varies as the ball rises and falls. What do these two observations indicate about the ball’s motion?

4. Hopefully you noticed that this second, up and down motion, is just like our ramp investigation. There was one minor but important difference. With our ramp, the ball moved first in a negative direction, then in a position direction. In this case the ball will travel in the positive, upward direction, then in the negative, downward direction. We’re going to do a repeat of the ramp investigation, this time with the billiard ball. Using what you’ve learned so far you should be able to draw predictions for all three graphs. Sketch your graphs, including the axes. Assume the upward direction is positive.

Lab 2.3 - Freefall 5 February 17, 2012 4a. Prediction: Position vs. time 4b. Prediction: Velocity vs. time 4c. Prediction: Acceleration vs. time

5. What will be the sign of the acceleration in this movie? (+ or -) OK, let’s try the Frame Flipper (FF). Click the Frame Flipper link to start it. This will open up a new window. You’ll recognize the image as the first frame of the movie. In this frame, the ball has just been set into motion. That is, it has just been released and is moving only under the influence of gravity and a negligible amount of air resistance. FF allows you to take position, time data of a moving object by clicking on it at each successive position. We’ll be interested in the vertical position only, so ignore the x values. In order to get actual measurements in meters we need to determine the scale of the image, that is, determine how many pixels tall our meter stick in the figure is (Steps 1 and 2). We then need to set the y = 0.0 m point (Step 3). Then we can start taking our data (Step 4 which repeats). You should be able to work this out by reading the directions on the screen. FF will automatically advance from Step 1 to Step 2 to Step 3 to Step 4 when you click on the screen. You’ll repeat Step 4 for as many frames as the ball takes to rise and fall, clicking the forward arrow each time to advance the movie. If you make an error, you can close the window and re-open it. Go ahead and collect all your data. To copy and paste your t and y data from FF to LP, you’ll need to copy it one column at a time from one to the other. In FF, click anywhere in the time data column. Right click and choose “Select All”, then right-click and select “Copy”. Open LP and paste the time data in the first column. Repeat for the y data. Not x! Autoscale your graph. You should have a nice down-opening parabola. (Hopefully that’s what you predicted.) In “Graph Options” turn on Point Protectors only. 6. Just as you did before, create your position, time, velocity, time, and acceleration, time graphs. Sketch these graphs. Note: 6c is very wiggly due to imprecision of frame-flipper data. My best effort is shown below. 6a. Observation: Position vs. time 6b. Observation: Velocity vs. time 6c. Observation: Acceleration vs. time

7. From the movie we can see that the ball is at rest (v = 0) at the top of its motion. How do the y,t & v,t graphs indicate this?

8. Does this mean that its acceleration is zero there? How do the v,t & a,t graphs answer this?

Lab 2.3 - Freefall 6 February 17, 2012 What we've found is that, in the absence of friction, vertical motion is motion at a constant acceleration. Note that the term vertical motion applies to either upward or downward motion. The magnitude of the acceleration due to gravity, g varies from location to location, but a typical value is 9.80 m/s2. Its direction is always downward. 9. You can probably get your best experimental value for this acceleration from the slope of your v, t graph. Record this below. Experimental value for g from v, t graph m/s2 Compare your experimental value to the accepted value by finding the percentage error between your experimental value and our accepted value. 10. Percentage error % Show calculations here.

Observations of Symmetry We noted earlier that the x,t graph was symmetrical. But we’ve also stated that these graphs model the motions we observe. What symmetries should we find in our models and our motions? 1. How does an object’s acceleration while rising compare to its acceleration while falling? Explain your answer using either the v,t graph or the a,t graph.

2. With this information and our five kinematics equations, we can solve problems with vertical motion. There are a couple of useful observations we can make from our data that will help with problem solving.

3. Using LP, click in the position, time graph and turn the tangent tool on. As you move your pointer along the graph a short tangent line will follow your cursor. At the bottom left of the graph you’ll see (time, position) values constantly updated. For some position, y on the left (ball rising) side of the graph, note the slope. Now repeat for a similar position, y on the right (ball falling) side of the graph. (You won't be able to get identical heights. Just get close.) How do these slopes compare. Repeat for another pair of matching heights. State your observations. (Refer to speed, not velocity.)

4. Now turn on the examine tool in LP. As you move your pointer across the position, time graph, the time to reach each y-position found in your data table appears in a box. Using this tool, find the time the ball took to rise between a point near y = .5m and y = 1.5m. s

Find the approximate time between the same positions on the way back down. s Pick another similar pair and repeat. (Again, this will just be approximate.) 5. State your observations.

Lab 2.3 - Freefall 7 February 17, 2012

Lab 2.3 - Freefall 8 February 17, 2012