Experiment E1510 Two Dimensional Collision

Figure 1. Two pucks collide on an air table. Discussion The Model Marking the Video Graphical Analysis

Discussion

Figure 1 shows two pucks colliding on an air table. An air table provides a cushion of air and reduces friction. Both the x and y directions, in this video, are real world horizontal. The video was taken using a tripod with the camera pointed down at the air table.

During the interaction (collision), the large puck applies a force on the small puck and the small puck applies a force on the large puck. If friction is sufficiently reduced on the air table and the air table is level, then these forces will be the only important forces. Therefore, the only important force on the large puck will come from the small puck and the only important force on the small puck will come from the large puck.

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The Model

See the E1400 series of video experiments for a more detailed discussion on Newton's Third Law.

Newton's Third Law:

If Object 1 applies a force on Object 2, then Object 2 applies an equal and opposite force on Object 1. These forces are called action reaction pairs and have these characteristics:

• The forces are on two different objects.
• Their magnitudes (absolute values) are the same.
• Their directions are opposite.

If Newton's Third Law is true and all other forces acting are not important, then:

F_1x = -F_2x and F_1y = -F_2y

Since force is the slope of a momentum-versus-time graph, this means that the slope of Object 1's momentum-versus-time graph should equal the negative slope of Object 2's momentum-versus-time graph during the interaction. Notice that these conditions are met in both the x and y directions.

Dp_1x = -Dp_2x and Dp_1y = -Dp_2y

Graphically, if one object's momentum changes by a certain value, then the other object's momentum will change by the negative of this value. Notice that these conditions are met in both the x and y directions.

Dp_Tx = 0 and Dp_Ty = 0

A plot of the total momentum versus time should be a horizontal line (zero slope). Again, notice that these conditions are met in both the x and y directions.

Finally, does the above discussion say the following?

• a_1x = -a_2x and a_1y = -a_2y the accelerations are equal and opposite in the x and y directions.
• Dv_1x = -Dv_2x and Dv_1y = -Dv_2y the velocity changes are equal and opposite in both the x and y directions.
• Dv_Tx = 0 and Dv_Ty = 0 the total velocity is conserved in both the x and y directions.

You should prove or disprove these last three statements while analyzing the video experiments.

Table 1 below shows how to find the magnitude and direction for acceleration, net force and impulse. You can add the additional subscripts 1 or 2 for the two different objects, but do not mix these subscripts in the same table. After finding the x and y components, the magnitude is the square root of the sum of the squares, and the direction is the inverse tangent of the y component divided by the x component in the correct quadrant.

Table 1. Two dimensional equations for acceleration, net force and impulse.

x direction	y direction	magnitude	direction
ax= Dvx/Dt	ay= Dvy/Dt	(ax2 + ay2)0.5	tan-1(ay/ax)
Fx= Dpx/Dt	Fy= Dpy/Dt	(Fx2 + Fy2)0.5	tan-1(Fy/Fx)
Dpx	Dpy	(Dpx2 + Dpy2)0.5	tan-1(Dpy/Dpx)

After finding the resultant acceleration, force and impulse on each object, you can determine if Newton's Third Law is true.

Remember:

Acceleration is the slope of a velocity-versus-time graph. In two dimensions, there may be two accelerations. You must find the acceleration in the x direction (a_x) and in the y direction (a_y).

Net force is the slope of a momentum-versus-time graph. In two dimensions, there may be two net forces. You must find the net forces in the x direction (F_x) and in the y direction (F_y).

Dp is the change in momentum. In two directions, you must find both Dp_x and Dp_y.

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Marking the Video

1. Play the video and observe the motion.
2. Mark the larger mass until the video does not step any more.
3. Click the Data Set 1 button and you will see its caption change to Data Set 2. Mark the smaller mass until the video does not step any more. These marks will be shown as green circles.
4. Select the Circles-Show Current menu item. This will display only the current frame's data circles. Now select Circles and then Vectors - Direction of a menu item. This will show the directions of the accelerations. Step through the video. Are the accelerations equal in magnitude and opposite in direction during the collision?
5. Select Circles and then Vectors - Direction of F_net menu item. This will show the directions of the net forces. Step through the video. During the collision, are the net forces equal in magnitude and opposite in direction?
6. Based on your observations, make rough sketches (no numbers, just the general shape) of the velocity and momentum versus time graphs. Mark the horizontal and vertical axes with the correct variables. You should verify if your sketches are correct when you view these graphs.

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Graphical Analysis

1. Press the GRAPHS Button and the Data Analysis Choices will appear. Pick Option 3. Click the Next button.
2. Direction of Motion? Pick both.
3. From the plots options, select velocity and momentum. Click the Next button.
4. Look at the velocity-versus-time graphs (there will be one for the x direction and one for the y direction). You will see three plots on each graph, one for object 1's velocity, one for object 2's velocity and one for their sum (the total velocity). Are the velocity changes equal and opposite for the two objects? Is the total velocity conserved? Are the objects' accelerations equal and opposite during the collision? Are the forces equal and opposite during the collision? Is Newton's Third Law true?
5. Look at the momentum-versus-time graphs (there will be one for the x direction and one for the y direction). You will see three plots on each graph, one for object 1's momentum, one for object 2's momentum and one for their sum (the total momentum). Are the impulses equal and opposite for the two objects? Is the total momentum conserved? Are the forces equal and opposite during the collision? Is Newton's Third Law true?
6. Create a table similar to Table 1 in the Model Section for each object. Find the magnitude and direction for each object's acceleration, net force and impulse. Which of these quantities are equal in magnitude and opposite in direction for the two objects?

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