Accelerated Motion

Acceleration is defined as a change in velocity during a time interval. The change in velocity can either be a change in speed, a change in direction of motion, or both. Mathematically, average acceleration is defined as change in velocity divided by change in time (Dv/Dt). Acceleration has a direction and a magnitude.  As we will soon learn, something with both direction and magnitude is called a vector quantity. For the special case of motion along a line, which you will explore in this experiment, the algebraic sign of the acceleration indicates the direction. 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 involve acceleration. Let’s 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 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

 LabPro interface ramp TI-83+  Graphing Calculator dynamics cart Vernier Motion Detector books DataMate program carpet square (or towel)

Pre-lab questions

Answer the first four questions for a car uniformly accelerating from rest for 10 seconds.

1.      How would the car’s velocity after four seconds compare to its velocity after two seconds? How would its velocity at ten seconds compare to the velocity at two seconds?

2.      Sketch a velocity vs. time graph of the car as it uniformly picks up speed. Describe in words what this graph means.

3.      Would the car cover more distance over the first two seconds of acceleration or the last two seconds?

4.      Sketch a graph of the distance vs. time as the car accelerates. Describe in words what this graph means.

5.      Sketch a velocity vs. time graph of a car as it slows down uniformly. Describe in words what this graph means.

6.      Sketch distance vs. time graph as a car slows down and finally stops. Describe in words what this graph means.

Procedure

1.      Place a few books under one end of the long board so that it forms about a 5° -10° 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 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.

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

Part I Speeding Up

5.      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.

6.      Select START to begin collecting data. After the Motion Detector starts to click, hold the cart for about one second, and then release it. Get your hand out of the Motion Detector path quickly.

7.      Press  to examine the distance vs. time graph. Repeat Steps 5 and 6 if your distance vs. time graph does not show areas 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, select MAIN SCREEN and return to step 5.

8.      Answer the Analysis questions for this part before proceeding to Part II.

Part II Slowing Down

In this part you will analyze the cart’s motion as it comes to a stop on the carpet.

9.      As you did before, hold the cart on the incline about 0.5 m from the Motion Detector.

10.    Select START to begin collecting data. After the Motion Detector starts to click, hold the cart for about 1 second, and then release it. Get your hand out of the Motion Detector path quickly.

11.    Press  to examine the distance vs. time graph. Repeat data collection if your distance vs. time graph does not show areas 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, select MAIN SCREEN and return to Step 9.

Data Table

Part I Speeding Up

 Time         (s) Velocity  (m/s) DVelocity  (m/s) DTime        (s) Average acceleration   (m/s2) Speeding up begins Speeding up ends

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

 Quadratic curve fit for distance data      (y = A*X2 + BX + C)

Part II Slowing Down

 Time         (s) Velocity  (m/s) DVelocity  (m/s) DTime        (s) Average acceleration   (m/s2) Slowing down begins Slowing down ends

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

 Quadratic curve fit for distance data      (y = A*X2 + 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. To do this, press  to return to the graph menu. Use the cursor keys to select VELOCITY, and press . 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.

a.    If you are not on the graph selection screen, if necessary navigate to the main screen and select GRAPH.

b.    Press  to select VELOCITY.

c.    Select SELECT REGION from the graph selection screen.

d.    Using the cursor keys, move the lower-bound cursor to the point when the cart first began to accelerate.

e.    Press  to record the lower bound.

f.      Using the cursor keys, move the upper-bound cursor to the point when the cart stopped accelerating uniformly.

g.    Press  to record the upper bound.

h.    After the selection is complete, graph selection screen will return. Press  to display the velocity graph. You will see the selected portion of your graph filling the width of the screen.

i.      Print or sketch this graph. Describe the graph in words.

5.      Both the distance and velocity 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.

a.    Return to the main screen by pressing , then selecting MAIN SCREEN.

b.    Select ANALYZE from the main screen.

c.    Select CURVE FIT from the ANALYZE OPTIONS.

d.    Select LINEAR (VELO VS TIME) from the CURVE FIT screen.

e.    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.

b.    Select CURVE FIT from the ANALYZE OPTIONS.

c.    Select QUAD (DIST VS TIME) from the CURVE FIT screen.

d.    Record the parameters of the curve fit in the Data Table.

e.    Press  to view the fitted curve with your data.

9.      How does the coefficient of the x2 term compare to the acceleration of the cart that you determined in Step 2?  How closely does the quadratic function fit the data?

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.

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.

a.    If you are not on the graph selection screen, if necessary navigate to the main screen and select GRAPH.

b.    Press  to select VELOCITY.

c.    Select SELECT REGION from the graph selection screen.

d.    Using the cursor keys, move the lower-bound cursor to the point when the cart first began to slow down.

e.    Press  to record the lower bound.

f.      Using the cursor keys, move the upper-bound cursor to the point when the cart stopped.

g.    Press  to record the upper bound.

h.    After the selection is complete, graph selection screen will return. Press  to display the velocity graph. You will see the selected portion of your graph filling the width of the screen.

i.      Print or sketch this graph.

j.      Describe the graph in words.

13.    To examine the distance vs. time graph during the negative acceleration:

b.    Use the cursor keys  and  to select DISTANCE.

c.    Press  to view the distance vs. time graph.

d.    Print or sketch this graph.

e.    Describe in words what the graph looks like.

14.    To fit a linear function to the velocity data:

a.    Press , and select MAIN SCREEN from the graph selection screen.

b.    Select ANALYZE from the main screen.

c.    Select CURVE FIT from the ANALYZE OPTIONS.

d.    Select LINEAR (VELO VS TIME) from the CURVE FIT menu.

e.    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.

b.    Select CURVE FIT from the ANALYZE OPTIONS.

c.    Select QUAD (DIST VS TIME) from the CURVE FIT screen.

d.    Record the parameters of the curve fit in the Data Table.

e.    Press  to view the fitted curve with your data.

18.    How does the coefficient of the x2 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 position-measurement scheme by putting the Motion Detector at the bottom of the ramp.

Extensions

1.   Determine if adding mass to the cart changes the acceleration of the cart on the ramp.

2.   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?