Objectives

• Collect distance, velocity, and acceleration data as a ball travels straight up and down.
• Analyze the distance vs. time, velocity vs. time, and acceleration vs. time graphs.
• Determine the best fit equations for the distance vs. time and velocity vs. time graphs.
• Determine the mean acceleration from the acceleration vs. time graph.

Introduction

When a juggler tosses a ball straight upward, the ball slows down until it reaches the top of its path. The ball then speeds up on its way back down. A graph of its velocity vs. time would show these changes. Is there a mathematical pattern to the changes in velocity? What is the accompanying pattern to the distance vs. time graph? What would the acceleration vs. time graph look like?

In this experiment, you will use a Motion Detector to collect distance, velocity, and acceleration data for a ball thrown straight upward. Analysis of the graphs of this motion will answer the questions asked above.

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Preliminary Questions

1. Think about the changes in motion the ball will undergo as it travels straight up and down. Make a sketch of a position versus time graph. Describe in words what parts of the graph is showing.

2. Make a sketch of the velocity versus time graph. Describe in words what parts of the graph is showing.

3. Make a sketch of the acceleration versus time graph. Describe in words what parts of the graph is showing.

4. On each of the three graphs, write the equation that describes the graph.
The equations are written in terms of the graph variables as:  x =...  v =...  a =...  as functions of time and acceleration)

Procedure

1. Sketch in your notes the experimental setup. Check the connection of the Motion Detector to the LabPro box and to the computer.

2. Prepare the computer for data collection by opening the LoggerPro software with the "MotionBallToss" experiment. Three graphs will be displayed: distance vs. time, velocity vs. time, and acceleration vs. time.

3. In this step, you will toss the ball straight up towards the Motion Detector with it also falling back down away from the motion detector. This step will require some practice. Hold the ball directly below the Motion Detector. Click "Collect" to begin data collection. When you will notice a clicking sound from the Motion Detector, wait a second and then toss the ball straight upward towards the detector. Keep your toss from hitting the detector by staying away by at least 0.5m (any closer and the motion detector will give faulty readings showing up as anomalous readings on your graphs). Smaller/lower tosses give the best results with the graphs scaled to fit the data.

4. Scale your three graphs for the best view of the ball toss region. Graph scaling is done by changing the scale end value by clicking on it. Adjust the x-axis scale to be the same for all three graphs with the y-axis of each graph adjusted for the best/largest view of the data.

5. Examine the distance vs. time graph. Repeat Step 3 until your distance vs. time graph shows an area of smoothly changing distance.

6. Sketch a copy in your notebook of all 3 motions graphs (as shown on the screen, keeping the relation between parts of the graphs the same).

7.  Repeat the ball toss experiment five more times. Each time, fit a straight line to the free-fall portion of the velocity graph and record the slope of that line. Average your six slopes to find a final value for your measurement of g. 

Analysis

1. The graphs you have recorded are fairly complex and it is important to identify different regions of each graph. Click the Examine button  and move the mouse across any graph to answer the following questions.

a) Identify the region when the ball was being tossed but still in your hands:

· Examine the velocity vs. time graph and identify this region. Label this on the graph.

· Examine the acceleration vs. time graph and identify the same region. Label this on the graph.

b) Identify the region where the ball is in free fall (defined as when gravity is the only force acting on the ball):

· Label the region on each graph where the ball was in free fall and moving upward.

· Label the region on each graph where the ball was in free fall and moving downward.

c) Determine the distance, velocity, and acceleration at specific points.

· On the velocity vs. time graph, decide where the ball had its maximum speed (magnitude of velocity), just as the ball was released. Mark the spot and record the value on the graph.

· On the distance vs. time graph, locate where the ball reaches its the maximum height during free fall. Mark the spot and record the value (distance from sensor) on the graph.

· What was the velocity of the ball at the top of its motion?

· What was the acceleration of the ball at the top of its motion?

2. The motion of an object in free fall is modeled by y = y₀ + v₀t + ½ a_yt², where y is the vertical position starting at  y₀, initial velocity is v₀, time is t, and a_y is acceleration due to gravity (9.8 m/s²)(*Note that due to the setup down is measured as the positive direction). This is a quadratic equation whose graph is a parabola. Your graph of distance vs. time should be parabolic. To fit a quadratic equation to your data, click and drag the mouse across the portion of the distance vs. time graph that is parabolic, highlighting the free-fall portion. Select "Analysis/Automatic Curve Fit" select quadratic fit from the list of models. Examine the fit of the curve to your data and click "OK" to return to the main graph. Determine g from your quadratic fit (related to coefficient of x²). How closely does this g compare to expected?

3. The graph of velocity vs. time should be linear. To fit a line to this data, click and drag the mouse across the free-fall region of the motion. Click the Regression button . How closely does the coefficient of the x term compare to the accepted value for g?

4. The graph of acceleration vs. time should appear to be more or less constant. Click and drag the mouse across the free-fall section of the motion and select "Analysis/Statistics". How closely does the mean acceleration value compare to the values of g found in Steps 2 and 3?

5. From the three graphs, determine your average value for g. Compare your value of g with the expected value using absolute difference and percent error. Write a sentence for each way of doing the comparison.

6. List some reasons why your values for the ball’s acceleration may be different from the accepted value for g.

Procedure 2

1. Do the ball toss experiment with a small steel ball. One would expect that air resistance (and buoyancy) would be less than with the larger soccer ball (leading to a better value). Compare acceleration determined of the steel ball toss and the soccer ball toss using percent difference (writing a statement using percent difference).