Experiment 9
Conservation of Linear Momentum
Objective:
To verify the law of conservation of linear momentum for head-on collisions of two masses
Equipment:
An air track with gliders, two sets of photo-gates (motion sensitive timers), additional weights, a mass scale, and a calculator
Theory:
If the centers of two billiard balls (moving toward each other) are exactly on the same line, Figure 1, their collision is a "head-on" collision, and after collision they tend to stay on the same line. If their centers are not exactly on the same line but are on two different lines (parallel or nonparallel), their collision is called "oblique collision" and after collision each take a different direction. In this experiment, the easier case of "head-on collision" will be examined in order to verify the law of conservation of linear momentum. Although on an air track, there are sliders (or gliders) instead of billiard balls, since the center of mass of both sliders move exactly on the same line, the collision is considered " head-on."
The Law of Conservation of Linear Momentum applied to masses M_{1} and M_{2} initially moving toward each other at velocities V_{1} and V_{2}, and finally (after collision) returning at velocities U_{1} and U_{2}_{ }may be written as follows:
M_{1}U_{1} + M_{2}U_{2} = M_{1}V_{1} + M_{2}V_{2} (1)
or,
Total Momentum After Collision = Total Momentum Before Collision
Elastic and Inelastic Collisions:
A collision during which some kinetic energy is lost is called "inelastic Collision." In reality, all collisions are inelastic and are associated with some K.E. loss no matter how small. A collision during which no K.E. is lost is considered "elastic" that means "perfectly elastic" and is ideal. The K.E. loss may be calculated as follows:
ΔK.E. = (Total K.E.)_{after collision } - (Total K.E.)_{before collision } =
[0.5M_{1}(U_{1})^{2} + 0.5M_{2}(U_{2})^{2}] - [0.5M_{1}(V_{1})^{2} + 0.5M_{2}(V_{2})^{2}]
Procedure:
1) Place the air track on an appropriate table. Connect the air pump to it and turn it on. Place two gliders (sliders) on it and use the leveling screws under its legs to level it. When leveled, the gliders should not move to either side if both are already at rest. This guarantees that there will be no change in the P.E. of the gliders when you make them move left or right.
2) Now, turn the air pump off. Do not slide the gliders when the air pump is off. This will keep the air track scratch-free.
3) Make sure that each glider has its two sticks inserted into its top holes near its top ends. These two sticks trigger the photo gate as the glider moves under it for the purpose of time measurement. Measure the mass of each glider (including its two rubber bands and the two sticks) using a laboratory scale. If they are exactly equal, you do not need to mark them; otherwise, mark them to make sure you know the mass of each. Let the left glider be M_{1} and the right glider M_{2} as indicated in Figure 2.
4) Turn the air track on and place the gliders on the air track, one at each end with their rubber bands facing each other. Place M1 on the left and M2 on the right as shown in Fig. 2. Each rubber band holder has an axle or a stem that gets horizontally inserted into a hole on the forehead of each glider. The holder can be turned about its horizontal stem to any angular position. Preferably, adjust the rubber bands angles such that they collide with each other at a 90-degree angle. This provides a smooth collision without any jumping of the collider (s) . Then practice sending the gliders toward each other by giving each a quick hit (an initial velocity) such that they meet somewhere near the middle of the air track. Two good points to put the photo gates at are at about 70cm and 130cm from say the left side. Left side is where M1 starts. Gently adjust the height of the photo gates by loosening the appropriate screws. Each time one of the sticks on top of a glider goes through the photo gate, the red light on top of that gate blinks, if, of course, the photo gate is already turned on. The closer the photo gates are the closer to the actual velocities before and after collision will be measured. The goal is to measure the velocities just before and just after collision. If photo gates are placed far from each other, they measure the initial velocities V_{1} and V_{2} too soon before collision and the final velocities U_{1} and U_{2} too late after collision allowing the photo gates to measure velocities that are not close enough to the actual velocities. That's why practicing the process a few times is important. If you can make gliders meet somewhere near the middle, then the 70cm and 130cm positioning of the gliders help better velocity measurements.
Time Measurements for the Calculation of Velocities (IMPORTANT)
5) Each time the two sticks on top of a glider go through a photo gate, the gate measures the time interval between the two generated pulses or events; therefore, each photo gate MUST be put in PULSE mode. Dividing the measured distance between the sticks of a glider (X) by the time it measures (t) results in a speed measurement. Fortunately, there is a good feature in this photo gate. It is its "MEMORY OPTION." If you turn the MEMORY OPTION on, the first time a glider goes through it, it records the elapsed time (t_{1}) for calculating a V. Do nothing and let the glider return after collision and go through that gate again. It has already measured (t_{2}) as well. What you read on it is (t_{1}). If you push the memory key to READ position, it shows you the total time (t_{1}+t_{2}). Subtract (t_{1}) from the total to find (t_{2}). (t_{2}) is the time for calculation of a U. With this feature, all you need to be good at is to make sure that the two gliders completely pass both gates before they collide. You may want to write down the times measured for the motion of M_{1} as (t_{11}) and (t_{12}) and the times measured for the motion of M_{2} as (t_{21}) and (t_{22}). The speeds will therefore be
V_{1} = X_{1} / t_{11 ; }U_{1} = X_{1} / t_{12} and
V_{2} = X_{2} / t_{21 ; }U_{2} = X_{2} / t_{22 }. Note that velocities must be plugged into equations and not just the speeds; in other words, do not forget the directions.
5) Apply the above procedure for the 4 cases shown in Table 1.
6) For each set of M_{1} and M_{2} in Table 1, plug V_{1} , U_{1}, V_{2} , and U_{2} in Equation (1) to see if the left side and right side become equal and if linear momentum is conserved.
7) For each case, calculate a %difference as well as the loss in the K.E. during collision. Record your results in Table 2.
Data:
Given:
Case | M_{1} | M_{2} |
1 | Glider +60.0grams* | Glider |
2 | Glider + 40.0grams* | Glider + 140.0grams* |
3 | Just the glider | Just the glider |
4 | Left glider at rest placed at the middle (V_{1} = 0) | Right glider put into motion toward the middle |
* The mass to be added must be split equally to both sides of the glider for symmetry |
Table 1
Measured:
Case | M_{1} _{ }(grams) |
V_{1} (cm/s) |
M_{2 }
(grams) |
V_{2} (cm/s) |
M_{1}V_{1}+M_{2}V_{2} _{ }(gram*cm/s) |
U_{1} (cm/s) |
U_{2} (cm/s) |
M_{1}U_{1}+M_{2}U_{2} _{ }(gram*cm/s) |
% diff on Total Momentum |
ΔK.E. Joules |
1 | ||||||||||
2 | ||||||||||
3 | ||||||||||
4 |
Comparison of the results:
Provide the percent difference formula used as well as the calculated percent difference in each case.
Conclusion:
State your conclusions of the experiment.
Discussion:
Provide a discussion if necessary.