One Dimension: Conservation of Linear Momentum Misaki, Christine, and Bonny

Elastic Collisions Elastic collisions: objects do NOT stick together after collision (the two general cases are shown below) Momentum conserved m 1 v i + m 2 v i = m 1 v f + m 2 v f Kinetic energy conversed ½ m 1 v i + ½ m 2 v i = ½ m 1 v f + ½m 2 v f Only under certain conditions: – Force of friction is negligible – Energy is not lost in the form of sound or heat If two objects have the SAME mass, all of the KE will be transferred to the other (assuming that the second object was initially at rest): If the two objects have DIFFERENT masses, a collision like this may happen:

Inelastic Collision Perfectly inelastic collision: colliding objects stick together after the collision – In reality, everything is NOT perfectly inelastic, but in physics problems done in class at this level, inelastic collisions implies that it is perfectly inelastic. Momentum conserved m 1 v i + m 2 v i = MV and M = m 1 + m 2 Kinetic energy NOT conserved Unlike elastic collisions, inelastic collisions can happen regardless of the two objects mass:

+ and - LoggerPro only allows the user to set ONE origin (shown with the yellow intersecting lines) Hovercraft 1’s initial position will be x = 0 (at the origin) Hovercraft 2’s initial position will be x = some negative number b/c it will be the displacement relative to origin – As it moves towards HC1, the HC2’s position will become less negative (the number will increase) This will also explain why the velocities of the two HCs are negative. Hovercraft 1 Hovercraft 2 Notes on using LoggerPro: Set origin (you can tilt the origin to fit your situation using the yellow dot on the right hand side as shown above) Use a meter stick or some sort of reference point for distance

Demonstration of 1D Collision B/c this demonstration was not done in ideal situations, there is a force of friction between the Hovercrafts and the floor. Although very small, there is some air resistance as well. In real situations, it is very hard to observe the conservation of linear momentum. Often, some of the KE will be lost.

Position vs. Time Graphs of Both Hovercrafts Graphs will be explained in detail in the following slides!

Hovercraft 1: Position vs. Time Slope of the best fit line is the velocity of HC1 Negative value b/c the origin was set on the right and the object was moving away from the origin towards the left Approximately at t=7.4, the velocity of hovercraft 1 becomes zero b/c all kinetic energy passed onto hovercraft 2 since they have the same mass.

Hovercraft 2: Position vs. Time Slope of the best fit line is velocity (v= m/s) – It is NOT perfectly linear b/c some errors were made when plotting the points with LoggerPro (human error). Velocity of HC is zero from t=0 to t=7.4 b/c the HC2 was at rest until HC1 hit HC2 at immediately after t=7.4

Hovercraft 1: Velocity vs. Time Slopes of best fit lines are the acceleration. Initially, the HC accelerates at the value of a=9.7 m/s 2 due to the force exerted on it by the initial push. Then, the HC starts to decelerate b/c of the force of friction that overcomes the initial force provided by the push. LoggerPro Notes These adjustments need to be made b/c of how the origin was set in this program. Negative acceleration values = HC is accelerating Positive acceleration values = HC is decelerating

Hovercraft 2: Velocity vs. Time Slopes of best fit lines are the acceleration. Initially, the HC accelerates at the value of a=9.7 m/s 2 due to the force exerted on it by HC1. Then, the HC2 starts to decelerate b/c of the force of friction that overcomes the initial force provided by the push.

Conservation of Momentum P initial = P final m 1 v i + m 2 v i = m 1 v f + m 2 v f m 1 =m 2 m 1 v i + m 2 v i = m 1 v f + m 2 v f ≈ ≈.6961 Loss of kinetic energy can be accounted for by the effects of the external forces. Results also follow a logical pattern: Hovercraft 1 has a higher velocity than the resulting velocity of Hovercraft 2. Some of the energy could have been translated into heat or sound at the time of the collision. **Human error when plotting the points also explain why the two numbers are not perfectly equal to each other. Using velocities immediately before and after the collision will yield the best results for the calculations.