Lab 15: Collisions in two dimensions

Collisions in Two Dimensions

I-Shou Lin, Kirk Paderes, Jorge Avalos

4/24/17

To record a two-dimensional collision with two balls and analyze if momentum and energy are conserved through graphical analysis

When collisions occur, they often last for just a small instant in time. However, during the collision, the objects that are being collided experience changes in both kinetic energy and momentum. In the case of a two-dimensional collision with two balls of different masses, a recording device can capture the motion of the two balls before, during, and after the collision. With the help of a logger pro, the position vs time graph for the two balls throughout the entire " journey" can be produced. Then, we verify if the momentum and kinetic energy is conserved in this collision. 

The laboratory procedure consisted of putting a recording device on the top of the stick over the glass table. Then, the balls collide with one another and then the recording is taken and uploaded onto logger pro for further analysis. 


( leveled glass table apparatus) 
Data Table

Experiment 1	Ball 1	Ball 2
Mass	0.005 kg	0.005 kg
Velocity, x ( initial)	0.1137 m/s	0.000 m/s
Velocity, x ( final)	0.04678 m/s	0.0494 m/s
Velocity, Y ( initial)	-0.0041 m/s	0.000 m/s
Velocity, Y ( final)	0.02755 m/s	-0.02964 m/s

experiment 2	Ball 1	Ball 2
Mass	0.005 kg	0.003 kg
Velocity, x ( initial)	0.1179 m/s	0.000 m/s
Velocity, x ( final)	0.05373 m/s	0.05131 m/s
Velocity, Y ( initial)	-0.002279 m/s	0.000 m/s
Velocity, Y ( final)	0.02042 m/s	-0.02275 m/s

( screenshot of experiment 1)

( screenshot of experiment 2) 

               ( Xcm and Ycm vs t, and Vxcm and Vycm vs t)

 ( Experiment 1)

 Xcm and Ycm vs t, and Vxcm and Vycm vs t)

( Experiment 2) 

Conservation of Momentum ( experiment 1)

                    X component

       m(1x1i)+m(2x2i) = m(1x1f)+m(2x2f)

       0.005 kg*0.1137 m/s +0.005 kg*0.00 m/s=0.005 kg*0.04678 m/s+0.005 kg * 0.0494 m/s

       0.1137 m/s+0.000 m/s = 0.04678 m/s+ 0.0494 m/s

        0.1137 m/s = 0.09618 m/s  

          not conserved, but close 

Y-Component

m(1y1i)+ m(2y2i)= m(1y2f)+ m (2y2f)

y1i+y2i= y1f+y2f

-0.0041 m/s+0.000 m/s= 0.02755 m/s+ (-0.02964 m/s)

-0.0041 m/s = -0.00209 m/s

 not conserved

Conservation of kinetic energy ( experiment 2) 

k.e.i=k.e.f

The first two graphs depict the position of the two balls before, during, and after collision in their components for experiment 1 and experiment 2. With the position of their components graphed, it makes the analyzing how the collision affected the position of the two balls. Then, the velocity of the two balls can be determined through the change in position divided by the elapsed time. The last two graphs show the position vs time graph and velocity vs time graph of the x and y components of the center of mass of the two system. The center of mass divided into the x and y components tells us if whether the collision is elastic or not by looking to see the slope of the position vs time graph for acceleration and by looking at the velocity of the center of mass vs time graph to see if there are changes in the slope of the velocity of the center of mass vs time graph. The position of the center of mass can be calculated through the center of mass formula. and the velocity of mass is found by taking the derivative of the center of mass formula with respect to time.

This lab was a fairly straightforward lab in terms of setup and data recording. However, analyzing the data for position vs time and then graphing these for analysis was more time consuming. The two main principles in this experiment are conservation of momentum and conservation of kinetic energy. In an ideal world, momentum and kinetic energy would both be conserved. The results show that momentum and kinetic energy aren't conserved. The momentum values before and after the collision are fairly close, but the kinetic energy values aren't. These may be due to the work done by non-conservative forces such as friction and other forms of energy such as sound energy. When the two balls collided, there was a noticeable sound. This energy came from the initial kinetic energy of the two balls. Therefore, the collision isn't an elastic collision. 

Lab 14: Impulse-Momentum Activity

Impulse-Momentum Lab 

I-Shou Lin, Kirk Paderes, Jorge Avalos

4/19/17

To measure, observe, and verify the impulse-momentum theorem

In the case of a cart on a track that's going to collide with an immobile surface, there's a force that acts on the cart which causes the cart to reverse its direction. However, since there're other net external forces that act on the cart during collision such as friction and air resistance, the collision is only nearly elastic. In the absence of all other net external forces, the collision is perfectly elastic, and the impulse momentum theorem is valid. We can verify that the impulse is equal to the change in momentum by plotting graphs of force versus time and velocity vs time, and then find the area under the curve of the graph of force versus time, and compare that integral value with the change in velocity multiplied by the mass. These two values should be equal or very close to each other. 

The entire lab is a collision experiment of nearly elastic and perfectly inelastic collisions. In the first experiment, we setup the rod clamped to a lab table with a cart attached to a rod and a track for the other cart with a force sensor mounted on that cart. It's setup so that the stopper of the moving cart hits the plunger of the stationary cart. Then, we ensured that the track is level, activated the motion detector, and calibrated the force probe. Finally, we collide the cart against the other cart several times until a good a set of graphs is obtained. Experiment 2 is just the same steps as experiment 1 except with several hundreds of grams of more masses. Experiment 3 is the completely inelastic collision in which the cart sticks to the wall. The cart attached to the rod has been replaced with a block of wood with a piece of clay attached to the front. The rubber stopper is replaced with a nail. We then collide the cart against the clay several times until we get the same initial velocity as the previous experiments. 

( Set up of experiment 1 with two carts)

( collision of experiment 3 in action)

Data Table

	Trial 1	Trial 2	Trial 3
Mass ( kg)	0.701	1.101	0.704
Velocity initial (m/s)	0.544	0.4593	0.500
Velocity final  ( m/s)	-0.395	-0.3255	0

Calculated Results/Graphs

( graph of position, velocity, and force versus time for experiment 1 with integral value)

( graph of position, velocity, and force versus time for experiment 2 with integral value)

( graph of position, velocity, and force vs time for experiment 3 with integral value)

	Experiment 1	Experiment 2	Experiment 3
Integral value (s*N)	0.8256	0.6318	0.3251

experiment 1 

Δp= p ( final)- p (initial)= m(v ( final)-v ( initial))= 0.701 kg* ( -0.395 m/s- 0.544 m/s)= - 0.658 kg*m/s

experiment 2 

Δp= p ( final)- p (initial)= m(v ( final)-v ( initial))= 1.101 kg ( -0.3255 m/s- 0.4593 m/s)= -0.864 kg*m/s 

experiment 3 

Δp= p ( final)- p (initial)= m(v ( final)-v ( initial))= 0.7011 kg * ( 0 m/s- 0.500 m/s)= -0.352 kg *m/s

All three experiments show the graphs of position, velocity, and force vs time. The position vs time graph shows us the movement of the cart relative to the motion detector. In addition, the slope of the graph is velocity. The velocity vs time graph shows the rate of movement before and after the collision when the velocity drops from positive to negative. The velocity is positive when the cart is moving towards the motion detector, and the velocity is negative when the cart is moving away from the motion detector. The abrupt drop in values corresponds to a high negative slope which is the time of collision. Finally, the force vs time graph is the key to verifying the impulse-momentum equation. The area under the graph of force vs time is equal to the impulse which is equal to the change in momentum as stated by the impulse-momentum equation. Therefore, if we take the integral of the graph of force vs time, we can get the impulse, and compare that value with the theoretical calculations for change in momentum.


Conclusion

In doing this experiment, we recorded data and graphed the position, velocity, and force vs time graphs on logger pro in order to analyze the impulse momentum theorem. Our graphs for all three experiments are consistent with theoretical predictions of what the general shape of the force vs time graph should look like before, during, and after the collision. The usual sources of error of friction and air resistance have a negligible effect on the motion of the cart due to its short time interval in reaching the end of the track and the minuscule interval of time of the collision. The calculated change in momentum of the cart didn't equal the measured impulse applied to it during the nearly elastic collision because of the inadequacy of the spring bumper and the execution of the experiment. The impulse and change in momentum were not equal to each other using more massive cart. I predict that that the impulse will be smaller than the impulse in the nearly elastic collision.The impulse and momentum will still be equal to each other. The force time curve for the nearly elastic collision stretches longer in time than the force time curve for the inelastic collision. The curve for the inelastic collision is very sharp and abrupt. This makes sense because there's only a small instant of time in which the force acts before the cart stops. Therefore, the curve should be high and thin, and it also means that the change in momentum won't be as great as compared to an elastic collision. In theory, the change in momentum should equal to the impulse or force multiplied by time. But, reality dictates that it's not the case due to human mishaps in conducting the experiment itself.

Lab 13: Magnetic Potential Energy Lab

Magnetic Potential Energy Lab 

I-Shou Lin, Kirk Paderes, Jorge Avalos 

4/ 17/17

The experiment is to verify that the principle of conservation of energy applies to all types of systems, and to find an equation that describes the magnetic potential energy.

Energy is present in the universe at a fixed constant. According to the principle of conservation of energy, energy can't be destroyed or created, only transferred from one form to another. In this lab, the two energies of interest are the kinetic energy of the cart and the magnetic potential energy in the magnetic field. When the cart slides one a friction-less surface with a magnet attached to one end and another magnet with the same polarity at the end of the track, the cart's kinetic energy is transformed into magnetic potential energy as the cart is momentarily at rest. Then, the magnetic potential energy is transformed back into kinetic energy as the cart speeds up. However, the magnetic potential energy isn't constant, but varies with the separation distance. Thus, in order to find the magnetic force, another force must be set equal to the magnetic force. This way, we could derive an equation for the magnetic force and the potential energy based on the other force. 

We first begin with the setup of a friction-less glider on an air track with a magnet attached to one end and another magnet of the same polarity at the end of the track. Then, we set the motion detector at the end of the air track and raise the track at various angles to determine the separation distance between the cart and track when the cart is in equilibrium. At this point, the magnetic force is equal to the gravitational force in the x-component parallel to the track. We do this for several angles and get several distances until we have enough data to plot force versus separation distance. The equation of the curve of that graph is the magnetic force. In order to verify conservation of energy, we attach an aluminum reflector to the cart, and then mass the cart. Since the motion detector isn't at the same position as the magnet at the end of the air track, a determination of the separation distance between the magnets and the distance the motion detector reads is necessary. We then push the cart from the far end of the track and record the data. Afterwards, we make a single graph of the kinetic and magnetic potential energy of the system as a function of time.



Measured Data

Separation distance (m)	Angle of Elevation	Mass of Cart ( kg)
0.0352	0.7	0.347
0.029	2.5	0347
0.0236	3.8	0.347
0.02	5.50	0.347
0.0182	6.8	0.347
0.0165	7.9	0.347
0.0152	9.0	0.347

Calculated Results 

F= A*r^n= 0.0001761*r^(-1.920 +/-0.08972)

F= mg*sine θ= 0.347 kg * 9.81 m/s^2 * sine ( 0.7)= 0.04159 newtons

Separation distance= distance motion detector reads - extra distance 

kinetic energy= 0.5 * mass*velocity ^2

kinetic energy= 0.5* 0.347 kg * (0.152 m/s )^2= 0.004 joules 

U(r)= ( A^(r+1)) /r+1

U(r)= ((0.001761 +/-6.409*10^-5)^ (0.02+1)/ 0.02+1= 0.00152 joules 

The first graph shows the inverse relationship between the separation distance and the repulsion magnetic force exerted by the two magnets on each other. Since there's no standard force equation for the force of magnetism, we have to derive the function. By plotting a graph of force vs separation distance and then finding a curve fit equation, a function for the unknown magnetic force arises. The second pair of graphs show the relationship between kinetic, potential, and total energy versus time and position. What verifies the principle of conservation of energy is the graph of kinetic, potential, and total energy as a function of time. That graph is in very close agreement with the theoretical graph that we did before the experiment.Just by visual inspection, we see that as the kinetic energy decreases, potential energy increases by the same amount, and so the total energy in the system is constant.

This magnetic conservation of energy lab turned out to be a very insightful and informative lab. It validated the universal principle of conservation of energy The graph of the various energies versus time depict almost perfectly that total mechanical energy in a system is conserved if external forces such as friction and air resistance can be ignored. What's most uncertain in the derivation of the magnetic potential energy function and also the force function is the uncertainty in the values of the constants A & n. These propagated uncertainty could have a significant effect on the value of the magnetic potential energy function because it's the integral of the force function. Overall, the results from the graphs show a successful demonstration of conservation of energy.

Lab 11: Work-Kinetic Energy Theorem

Work-Kinetic Energy Theorem 

I-Shou Lin, May Soe Moe, Roya Bijanara

4/10/17

The purpose of this lab is to investigate and examine the work-kinetic energy theorem and how the work done is equal to the change in kinetic energy of an object. 

Energy is all around us, and there's work to be done everywhere ( no pun intended). The work-kinetic energy theorem relates energy to work by stating that the work done on an object causes a change in the object's kinetic energy. In order to compare work done and the change in kinetic energy, graphs of force vs distance and kinetic energy vs position are needed. The area under the curve of the force vs distance graph is the work done on the object. The area which is equal to the integral, can then be compared to the value of the kinetic energy at the corresponding points. 

Experiment 1: 
we set up the track, cart, motion detector, force probe, pulley, cart stop and hanging mass. The force sensor was zeroed first, and the 500 grams of mass was added to the cart. Then, we hung a 50 grams of mass from the end of the string, pulled the cart back and then released the cart to get the data. The idea is that the work done on the cart by the tension force in the string should equal the kinetic energy gained by the cart at any point during the acceleration of the cart.

Experiment 2: 
We measured the work done by a spring on the cart when the spring is stretched. Set up the ramp, cart, motion detector, force probe, and spring. Then, we zeroed the force probe and the motion detector so that toward the detector is the positive direction. In order to measure the work done by a constant force, a pull is required on the cart as it moves slowly to 0.6 m of stretched position.

Experiment 3: 
Part 3 of the experiment uses the same setup as part 2. However, this time we will measure the work done by a non-constant spring force on the cart. We measured the mass of the cart and entered a formula that allows us to calculate the kinetic energy of the cart at any point. We zeroed the force probe and the motion detector to be zero at the starting position. We pulled the cart so that the spring is stretched about 0.6 m from the equilibrium position. The cart is released, we begin graphing.

Experiment 4: 
In this part we justr watched a movie of a professor using a mchine to pull back on a large rubber band. The graph of force vs position is shown in the video.


Data Table

part 1:

	Integral value ( joules)	Kinetic energy ( joules)
Area 1 of the graph	0.1562	0.195
Area 2 of the graph	0.2755	0.232

part 2: 

Spring constant ( N/m)	Work done in stretching the spring ( joules)
3.511	0.21585

part 3: 

	Trial 1	Trial 2
Mass of the cart ( kg)	0.549	0.549
Δ kinetic energy ( joules)	0.109	0.150
Work done ( joules)	0.1287	0.1775

part 4: 

Mass of the cart ( kg)	Δ X photo-gate ( cm)	Δ T photo-gate ( milliseconds)
4.3	15	45

Calculated Results

part 4 of the experiment: 

k.e.( final)=0.5 * mass* velocity^2 ( final)= 0.5 * 4.3 kg* (0.15 m/0.045 s)^2= 24 joules 

For experiment 1 of the lab, the graphs depict force vs position, position vs time,  and velocity vs time, These graphs depict the relationship between work and the change in kinetic energy of the cart as well as the velocity and acceleration of the cart. The area under the curve of the force vs position graph shows the work done by the tension force in the string because work is force * distance. An integral is used because the force isn't constant. Velocity is the ratio of position divided by time, and acceleration is the ratio of velocity divided time.

For experiment 2, the two graphs depict the work done by a spring on a cart when the force is constant. Again, the area under the curve of the graph is equal to the work done by the spring because work is force multiplied by distance. In this case simple algebra is all that's needed to calculate the work done due to the force being constant. In addition, the last graph shows the work done by a constant spring force with a linear fit so that the spring constant can be determined.

For experiment 3, the five graphs transmit the work done by a spring force on a cart when the force isn't constant. Again, the area under the curve graph of the curve is equal to the work done by the spring because work is equal to force multiplied by distance. This time, an integral function is required because the force is not constant. In addition, the last 3 graphs of the 5 graphs analyzes the work done through the area under the curve and the change in kinetic energy of the cart. These two values should be equal in theory according to the work-kinetic energy theorem.

In conclusion, this lab was divided into four parts with the first three parts requiring collecting data to be graphed and then analyzing that data. For us it was a particularly confusing and unpleasant laboratory experience because of the number of technical functions involved in the computer software and graphing. We made many mistakes and had to perform each experiment many times. But, in the end we were able to produce good data and good results. Part four of the lab is just watching a video and then taking data in order to find the work done and the final kinetic energy. With respect to the comparison of the data, the results for experiment 1 for the integral and kinetic energy are within 20% of each other. Since they're fairly close, the original work-kinetic energy theorem holds. The work energy principle state that the work done on the cart-spring system is equal to the change in the cart-spring system's kinetic energy. And the work done on the cart and the change in the cart's kinetic energy are off by 16-17%. For the last experiment of the lab, the final kinetic energy of the cart is almost equal to the work done by the professor. The work done by the professor is 23.25 joules, and the cart's final kinetic energy is 23.9 joules. That is a percent difference of 2.79. These two values won't be the same because of the presence of friction and air resistance when the cart was accelerating through the two photo-gates. Also, the graph of force vs position isn't entirely consistent, and this would cause the calculations for the work done by the professor to be subject to errors and propagated uncertainties.