Accelerated Motion
An acceleration is a change in velocity during a time interval. The change in velocity can either be a change in speed or a change in direction of motion. Mathematically, average acceleration is defined as change in velocity divided by change in time (Dv/Dt). Since acceleration is a vector quantity, it has a direction and a magnitude. For the special case of motion along a line ( one dimensional motion ), which you will explore in this experiment, the direction is indicated by the algebraic sign of the acceleration. That is, the acceleration can either be positive (in the direction of increasing positive distances from the origin) or negative (in the direction of decreasing positive distances from the origin). Most of the motions you experience or even notice involve acceleration. Lets consider a car traveling in increasing positive distances from a stop sign. While riding in a car that leaves from the stop sign, you undergo a positive acceleration. Later, as the car slows to a stop, you experience a negative acceleration. When a car picks up speed constantly, it undergoes uniform acceleration. In this experiment, you will collect and analyze distance and velocity data for a cart as it accelerates down a ramp and then rolls to a stop.
Figure 1
objectives
· Use a Motion Detector or Ranger to collect distance and velocity data as a cart accelerates down an incline and then rolls to a stop.
· Analyze graphs of distance vs. time and velocity vs. time for accelerated motion.
· Determine the mathematical relationship between the velocity and time for this motion.
· Determine the mathematical relationship between the distance and time for this motion.
Materials
TI82, 83, 86, 89, 92, or 92 Plus 
ramp 
CBL2 System * 
dynamics cart 
Vernier Motion Detector * 
books 
program loaded on calculator 
carpet square 
· or alternate setup of a ranger and the use of the software program graphical analysis
Procedure
1. Place two or three books under one end of a 1.2 3 m long board or track so that it forms about a 5° angle with the horizontal. Place the carpet square at the bottom of the ramp so that the cart will roll off the ramp onto the carpet.
2. Place the Motion Detector at the top of an incline.
3. Connect the Vernier Motion Detector to the port of the CBL unit. Use the black link cable to connect the CBL unit to the calculator. Firmly press in the cable ends.
4. Set up the calculator and CBL for the Motion Detector. Start the
program and proceed to the .· Select from the .
· Select as the number of probes.
· Select from the menu.
Part I Speeding Up
5. Set up the calculator and CBL for data collection.
· Select COLLECT DATA from the MAIN MENU.
· Select TIME GRAPH from the DATA COLLECTION menu.
· Enter 0.05 as the time between samples, in seconds.
· Enter 80 as the number of samples (the CBL will collect data for about 4 seconds).
· Press , then select USE TIME SETUP to continue. If you want to change the sample time or sample number, select instead.
6. Hold the cart on the incline about one meter from the bottom of the ramp and at least 0.5 m from the Motion Detector.
7. Press to begin collecting data. After the Motion Detector starts to click, hold the cart for about one second, then release it. Get your hand out of the Motion Detector path quickly.
8. Select from menu. Examine the distance vs. time graph. Repeat Steps 6 and 7 if your distance vs. time graph does not show areas of smoothly changing distance. Check with your teacher if you are not sure whether you need to repeat the data collection. To repeat data collection, press to return to the menu; select from the menu.
9. Answer the Analysis questions for this Part I before proceeding to Part II.
Part II Slowing Down
10. In this part you will analyze the carts motion as it comes to a stop on the carpet. Set up the calculator and CBL for a second run.
· Select COLLECT DATA from the MAIN MENU.
· Select TIME GRAPH from the DATA COLLECTION menu.
· Enter 0.05 as the time between samples, in seconds.
· Enter 80 as the number of samples (the CBL will collect data for about 4 seconds).
· Press , then select USE TIME SETUP to continue. If you want to change the sample time or sample number, select instead.
11. As you did before, hold the cart on the incline about 0.5 m from the Motion Detector.
12. Press to begin collecting data. After the Motion Detector starts to click, hold the cart for about 1 second, then release it. Get your hand out of the Motion Detector path quickly.
13. Select
from menu. Examine the distance vs. time graph. Repeat Steps 11 and 12 if your distance vs. time graph does not show areas of smoothly changing distance. Check with your teacher if you are not sure whether you need to repeat the data collection. To repeat data collection, press to return to the menu; select from the menu.14. Answer the remaining Analysis questions.
Data Table
Part I Speeding Up

Time (s) 
Velocity (m/s) 
DVelocity (m/s) 
DTime (s) 
Average acceleration (m/s^{2}) 
Speeding up begins 





Speeding up ends 





Linear curve fit for velocity data (y = A*X + B) 

Quadratic curve fit for distance data (y = A*X^{2} + BX + C) 

Part II Slowing Down

Time (s) 
Velocity (m/s) 
DVelocity (m/s) 
DTime (s) 
Average acceleration (m/s^{2}) 
Slowing down begins 





Slowing down ends 





Linear curve fit for velocity data (y = A*X + B) 

Quadratic curve fit for distance data (y = A*X^{2} + BX + C) 

Analysis
Part I Speeding Up
1. Either print or sketch the distance vs. time graph. The graph you have recorded contains regions for each part of the motion. It is important to identify these regions. Record your answers directly on the printed or sketched graph.
a) Examine the distance vs. time graph and identify when the cart was initially at rest on the ramp. Label this region on the graph.
b) Identify when the cart was accelerating down the ramp. Label this region on the graph.
c) Identify when the cart was slowing to a stop. Label this region on the graph.
d) Is the cart moving in the direction of increasing or decreasing distance from the origin as it rolls down the ramp? How can you tell?
2. View the velocity vs. time graph by pressing to return to the menu and selecting . Either print or sketch the graph. The graph you have recorded contains regions for each part of the motion. It is important to identify these regions. Record your answers directly on the printed or sketched graph.
a) Examine the velocity vs. time graph and identify when the cart was initially at rest on the ramp. Label this region on the graph.
b) Identify when the cart was accelerating down the ramp. Label this region on the graph.
c) Identify when the cart was slowing to a stop. Label this region on the graph.
3. Determine the acceleration of the cart on the ramp using the velocity graph. Use the cursor keys on the velocity vs. time graph to read numeric values.
a) On the graph, locate when the cart began to accelerate down the ramp. Record the beginning time and velocity in the Data Table.
b) Use the cursor keys to determine when the cart stopped its uniform acceleration. Record the ending time and velocity in the Data Table.
c) Calculate the change in velocity (D velocity) and the corresponding change in time (D time) and record your results in the Data Table.
d) Calculate the acceleration and record your results in the Data Table.
4. To examine the positive acceleration more closely, you need to first remove the data that do not correspond to the cart freely rolling down the ramp.
· Proceed to the .
· Select from the .
· Select from the .
· Select from the menu.
· Using the cursor keys, move the lowerbound cursor to the point when the cart first began to accelerate.
· Press to record the lower bound.
· Using the cursor keys, move the upperbound cursor to the point when the cart stopped accelerating uniformly.
· Press to record the upper bound.
· After the selection is complete, the menu will appear. Select from the menu. You will see the selected portion of your graph filling the width of the screen.
· Print or sketch this graph.
· Describe the graph in words.
( This next section can be done more easily using the software program Graphical analysis )
5. Each of these graphs can be modeled with a function. The graph of velocity vs. time should be linear. The calculator can fit a linear function to these data.
· Proceed to the .
· Select from the .
· Select from the .
· Select L1, L5 from the menu.
· Record the parameters of the linear curve fit in the Data Table.
6. How closely does the coefficient of the x term in Step 5 compare to the acceleration you calculated in Step 3?
7. Press to view the fitted curve with your data.
8. Next you can fit a quadratic function to the distance vs. time data.
· Proceed to the .
· Select from the .
· Select from the .
· Select L1, L4 from the menu.
· Record the parameters of the curve fit in the Data Table.
· Press to view the fitted curve with your data.
9. How does the coefficient of the x^{2} term compare to the acceleration of the cart that you determined in Step 2? How closely does the quadratic function fit the data?
10. Return to Step 10 of the Procedure.
Part II Slowing Down
11. Determine the acceleration of the cart on the carpet. Use the cursor keys on the velocity vs. time graph to read numeric values.
a) On the graph, locate when the cart began to slow down on the carpet. Record the beginning time and velocity in the Data Table.
b) Locate when the cart was about to stop. Record this time and velocity in the Data Table.
c) Calculate the change in velocity (D velocity) and the corresponding change in time (D time) and record your results in the Data Table.
d) Calculate the acceleration and record your results in the Data Table.
( This next section can be done more easily using the software program Graphical analysis )
12. To examine the negative acceleration more closely, you need to first remove data that do not correspond to the cart rolling on the carpet.
· Proceed to the .
· Select from the .
· Select from the .
· Select from the menu.
· Using the cursor keys to move the lower bound cursor to the point when the cart first began to slow down.
· Press to record the lower bound.
· Using the cursor keys move the upper bound cursor to the point when the cart stopped.
· Press to record the upper bound.
· Select from the menu. You will see the selected portion of your graph filling the width of the screen.
· Print or sketch this graph.
· Describe the graph in words.
13. To examine the distance vs. time graph during the negative acceleration:
· Select from the menu.
· Print or sketch this graph.
· Describe in words what the graph looks like.
14. To fit a linear function to the velocity data:
· Proceed to the .
· Select from the .
· Select from the .
· Select L1, L5 from the menu.
· Record the parameters of the linear curve fit in the Data Table.
15. How closely does the coefficient of the x term in Step 14 compare to the acceleration you calculated in Step 2? How closely does a linear function fit the data?
16. Press to view the fitted curve with your data.
17. Next you can fit a parabola to the distance vs. time data.
· Proceed to the .
· Select from the .
· Select from the .
· Select L1, L4 from the menu.
· Record the parameters of the curve fit in the Data Table.
· Press to view the fitted curve with your data.
18. How does the coefficient of the x^{2} term compare to the acceleration of the cart? How closely does the quadratic function fit the data?
19. Compare the accelerations in Part I and Part II. Which was larger?
20. If motion down the ramp had corresponded to decreasing positive distances to the origin (3, 2.5, 2, 1.5 ), how would the signs of the velocity and the acceleration be affected? You could obtain this different positionmeasurement scheme by putting the Motion Detector at the bottom of the ramp.
Extensions ( required )
1. Instead of releasing the cart from rest, give it a quick push down the ramp. Does this change the acceleration you measure after the push is complete and before the cart hits the carpet?