Physics Laboratory 7

Conservation of linear momentum

Objectives:

In this lab students will will analyze several video clips to explore conservation of linear momentum in collisions between two gliders on an air track. Students will measure the velocity of the two gliders before and after a collision. They will find the momentum and the kinetic energy of each glider and check if the total momentum and the kinetic energy of both gliders is conserved.

The momentum of an object is the product of its mass and its velocity. The total momentum is conserved in collisions of isolated objects. If no external forces act on two interacting objects, then the sum of the momenta of the two objects prior to a collision equals the sum of the momenta after the collision.

m₁v₁ + m₂v₂ = m₁v₁’ + m₂v₂’

If no external forces act along a particular direction, then the component of the total momentum in that direction is conserved.

If forces acting in the horizontal direction (such as friction) can be ignored in the experiments of this lab, then the sum of the momenta of the two gliders prior to a collision should be the same as the sum of the momenta of the gliders after the collision.

If the collision of the two gliders is elastic, then kinetic energy is also conserved.

Procedure:

Part I (Coefficient of restitution)

In order for the gliders to bounce off each other after a collision, the gliders are fitted with rubber-band bumpers. If the coefficient of restitution of a bumper would be one, then the collision would be elastic. Unfortunately, the coefficient of restitution is less than one, and we do not have a perfectly elastic collision. The coefficient of restitution of a rubber-band bumper depends on the physical condition of the rubber band. Students will measure the coefficient of restitution of one particular bumper to obtain a typical value.

Choose the coll_11.avi video clip. To play the video clip or to step through it frame-by-frame click the "Begin" button. The "Video Analysis" web page will open. You can toggle between the current page and the "Video Analysis" page by pressing Alt-Tab.

For instructions on how to use "Video Analysis on the Web" refer to a previous exercise.

In the video clip the glider collides with a bumper rigidly attached to the end of the track. Students will measure the speed of the glider before and after the collision. The will determine the coefficient of restitution = (outgoing speed)/(incoming speed).

In the setup window choose to track the x-coordinate of one object. To find the velocity of the cart before and after the collision with the bumper start taking data. Determine the position of the cart as a function of time by stepping through the video clip frame-by-frame and by reading the time and the position coordinates of the cart off each frame. Calibrate your data and construct an Excel spreadsheet with columns for time and position.

Plot the position of the cart as a function of time. Use Excels regression function to find the slopes the position versus time graph before and after the collision with the bumper. The slopes yield the velocities of the cart before and after the collision.

Determine the coefficient of restitution for the bumpers within some estimated uncertainty. The coefficient of restitution is the ratio of the speed after a collision over the speed before the collision. (Use the standard error in the X Variable (slope) given by Excel's regression function to estimated the uncertainty in the slope and therefore in the velocity.)

Construct a new "Results" spreadsheet as shown below and enter the coefficient of restitution and your estimated uncertainty.

Part II (Elastic Collisions)

Choose the coll_21.avi video clip. In this video clip two carts collide and bounce off each other. In the setup window choose to track the x-coordinate of two objects. To find the velocity of the carts before and after the collision start taking data. Determine the positions of the carts as a function of time by stepping through the video clip frame-by-frame and by reading the time and the position coordinates of the carts off each frame. Calibrate your data and construct an Excel spreadsheet with columns for time and positions.
Let the velocity be positive if the cart is moving towards the right and negative if it is moving towards the left. Make sure that you not only record the magnitude but also the direction of the velocity.

Plot the positions of the carts as a function of time. Use Excel's regression function or the trendline to find the slopes the positions versus time graphs before and after the collision. The slopes yield the velocities of the carts before and after the collision. (See this linked spreadsheet (sheet 1 and sheet 2) for a guide on how to proceed.)

Enter the masses and the initial and final velocities of the carts into your "Results" spreadsheet under "Elastic Collision".

Part III (Inelastic Collisions)

	Choose the coll_31.avi video clip. In this video clip two carts fitted with clay bumpers collide and stick together. In the setup window choose to track the x-coordinate of two objects. To find the velocity of the carts before and after the collision start taking data. Determine the positions of the carts as a function of time by stepping through the video clip frame-by-frame and by reading the time and the position coordinates of the carts off each frame. Calibrate your data and construct an Excel spreadsheet with columns for time and positions. Let the velocity be positive if the cart is moving towards the right and negative if it is moving towards the left. Make sure that you not only record the magnitude but also the direction of the velocity.
	Plot the positions of the carts as a function of time. Use Excel's regression function or the trendline to find the slopes the positions versus time graphs before and after the collision. The slopes yield the velocities of the carts before and after the collision.
	Enter the masses and the initial and final velocities of the carts into your "Results" spreadsheet under "Inelastic Collision"

Data Analysis:

Extend your "Results" spreadsheet. For the elastic and inelastic collision calculate:

	p_1i=m₁v_1i
	p_2i=m₂v_2i
	p_i=p_1i+p_2i
	p_1f=m₁v_1f
	p_2f=m₂v_2f
	p_f=p_1f+p_2f
	K_1i=(m₁v_1i²)/2
	K_2i=(m₂v_2i²)/2
	K_i=K_1i+K_2i
	K_1f=(m₁v_1f²)/2
	K_2f=(m₂v_2f²)/2
	K_f=K_1f+K_2f

Open Microsoft Word and prepare a report using the template shown below.

Name:
E-mail address:

Laboratory 7 Report

	In a few sentences summarize the experiment.
	Show your "Results" spreadsheet.

Answer the following questions in full sentences.

	Is momentum conserved in your elastic collision experiments? If not, what is the percentage difference between p_i and p_f.
	Is kinetic energy conserved in your elastic collision experiments? If not, what is the percentage difference between K_i and K_f.
	We may write 100(K_i-K_f)/K_i=100(v_i²-v_f²)/v_i²=100(1-v_f²/v_i²). What is the percentage difference between K_i and K_f that you would expect using your measured value of the coefficient of restitution of a typical bumper?
	Is momentum conserved in your inelastic collision experiments? If not, what is the percentage difference between p_i and p_f?
	Is kinetic energy conserved in your inelastic collision experiments? If not, what is the percentage difference between K_i and K_f.

Comment on your result. Did your experiment reproduce the expected results? If not, speculate on the reasons for any discrepancies.