Theory

Newton's second law can be written in a more general form as 

where is the momentum of system of N objects and is the net external force on the system. This relationship says that the rate at which a system's momentum changes is equal to the net external force exerted on it.

If there is no net external force exerted on a system, then the momentum is constant and we say that momentum is conserved. Under these circumstances if there is some interaction between the parts of a system, the momentum of the individual parts may change but the total momentum is the same.

When objects collide with each other, the forces between them can be quite large while the collision itself happens over a brief time interval. The influence of any external forces that may also be acting during the collision are therefore limited and so collisions are typically a great place to apply the principle of conservation of momentum even if it is just an approximation.

Initial momentum of the system is

where velocities can be positive and negative depending on the direction of motion. After the collision, the objects may stick together, inelastic collision, or bounce off each other, elastic collision.

The final momentum of the system for an elastic collision is:

And the linear momentum conservation law is:

The final momentum of the system for an inelastic collision is:

And the linear momentum conservation law is:

Procedure (Elastic/Inelastic Collisions)

The lab uses the simulation included below:

Collision Lab

Select "Intro", "Velocity" and "Values" "More Data" and "Slow".

Select 100% Elastic.

To adjust the value of masses, click on the white boxes under "More Data" and "Mass (kg) " corresponding to objects 1 and 2 and select the values 2.5" and "0.5" for objects 1 and 2 respectively, followed by "Enter".

m₁=2.5 kg and m₂=0.5kg

To adjust the values of velocities, click on the white boxes corresponding to objects 1 and 2 and select the value followed by "Enter"

v_1i=0m/s and v_2i=-0.5m/s.

Play "Play" and then "Pause" to check if your screen looks like the picture below. The values are entered in the tables 1-4 along with total momentum and kinetic energy calculations.

In the simulation, positive direction is to the right. Notice you must select the "-"if the object is moving to the left.

Linear momentum will be shown in absolute value next to the ball, but it will have the correct + or - value in the" More Data" table.

Determine velocities before and after collision and type them in the tables.

Place the objects at a reasonable distance such as the collision is happening on the screen and use the "Refresh" button anytime you want to start over.

Be careful about the velocities' signs and keep track which object is "1" (green) and which one is "2" (pink).

You can change the velocity by typing in the white box or by modifying the magnitude and direction of the arrow.

Play "Pause" just before collision and then again after collision, to record the values of velocities in the tables.

In runs 2- 4 you must type in.5 the values of the initial velocities' values and record carefully the velocities' values and signs after collisions. The + and - are indicating directions.

Table 1 Linear Momentum in Perfect Elastic Collisions

Run #	m1green (kg)	m2pink (kg)	v1i green (before) (m/s)	v2i pink (before) (m/s)	v1f green (after) (m/s)	v2f pink (after) (m/s)	Ptotal (initial) (kgm/s)	Ptotal (final) (kgm/s)
1	1.5	1.5	0.5	0	0	0.5	0.75	0.75
2	1.5	1.5	0.5	-0.5
3	3	1.5	0.5	0
4	1.5	3	0.5	0

INCLUDE YOUR CALCULATIONS FOR ONE OF THE RUNS

Table 2 Kinetic Energy in Perfect Elastic Collisions

Run #	m1green (kg)	m2pink (kg)	v1i green (before) (m/s)	v2i pink (before) (m/s)	v1f green (after) (m/s)	v2f pink (after) (m/s)	Ktotal (intial) (J)	Ktotal (final) (J)
1	1.5	1.5	0.5	0	0	0.5	0.19	0.19
2	1.5	1.5	0.5	-0.5
3	3	1.5	0.5	0
4	1.5	3	0.5	0

INCLUDE YOUR CALCULATIONS FOR ONE OF THE RUNS

Table 3 Linear Momentum in Perfect Inelastic Collisions

Run #	m1green (kg)	m2pink (kg)	v1i green (before) (m/s)	v2i pink (before) (m/s)	vf (m/s)	Ptotal (intial) (kgm/s)	Ptotal (final) (kgm/s)
1	1.5	1.5	0.5	0	0.25	0.75	0.75
2	1.5	1.5	0.5	-0.5
3	3	1.5	0.5	0
4	1.5	3	0.5	0

INCLUDE YOUR CALCULATIONS JUST FOR ONE OF THE RUNS

Table 4 Kinetic Energy in Perfect Inelastic Collisions

Run #	m1green (kg)	m2pink (kg)	v1i green (before) (m/s)	v2i pink (before) (m/s)	v1f (after) (m/s)	Ktotal (intial) (J)	Ktotal (final) (J)
1	1.5	1.5	0.5	0	0.25	0.19	0.09
2	1.5	1.5	0.5	-0.5
3	3	1.5	0.5	0
4	1.5	3	0.5	0

INCLUDE YOUR CALCULATIONS JUST FOR ONE OF THE RUNS

Questions

1. Define an elastic collision and indicate the conservation laws for it. Are the results of your tables 1 and 2 in agreement with these conservation laws?

1. Define an inelastic collision and indicate the conservation laws for it. Are the results of your tables 3 and 4 in agreement with these conservation laws?