Momentum of Inertia of a Triangular Plate

Purpose:
To use all of the concepts of moment of inertia to predict and make a reasonable experiment.

Equipment Used:

The device is a series of two wheels one on top of the other that has air flow through it and creates a rotating system with a negligible amount of friction. The rotating system has a sensor that reads the rotation of the wheels, both the top and the bottom, since the two wheels can be made to either spin independently or together as a single system. The hanging mass is on a pulley that has a similar design, allowing air to flow through it to create a negligible amount of friction. The black triangular plate is held from the center of mass and is the object that's moment of inertia will be calculated both theoretically and experimentally.

Data Collected:
The moment of inertia of an object is based on 3 things, the mass of the object, the axis of rotation, and the orientation and distance of the object from the axis of rotation. The triangular plate is treated as a thin plate.
The mass of the triangle = .456kg
Length of the base of triangle = .10m
Length of the height of triangle = .15m
The mass of the hanging object = .025g
Diameter of disk that is connected to hanging mass = .05m
In order to experimentally calculate the moment of inertia with the triangular plate in either position, the moment of inertia of the system must be calculated first. So a hanging mass is let loose and the angular acceleration of all three are recorded.
Angular accelerations
No plate = 1.337rad/s^2
Plate with base extended = 1.041rad/s^2
Plate with height extended = .989rad/s^2

Calculations:

The theoretical moment of inertia of the triangular plate in both positions are calculated first. The moments of inertia are found at the edges of rotation, so along the height or base and then the parallel axis theorem is used to move the center of mass to the center of the plate.

Using the angular accelerations, the numerical value for the moment of inertia can be calculated using the angular acceleration. Once the moment of inertia of the system without the plate is calculated, then the plate can be introduced to the system and in both positions it can be calculated.















The theoretical value for the moment of inertia rotated with the base out = .0019
The experimental value for the moment of inertia rotated with the base out = .0013
The theoretical value for the moment of inertia rotated with the height out = .0022
The experimental value for the moment of inertia rotated with the height out = .0016

Conclusion:
The predictions for the moment of inertia seemed to be over the value that was experimentally calculated. The likely source for the error was that the moment of inertia couldn't calculate for the hole in the center that is used to hold the triangle to the system.

Sincerely,

Swaggy C

Conservation of Momentum

Purpose:
The purpose of the experiment is to experimentally prove that momentum is conserved in a collision.

Equipment Used:

The picture is of an instrument that records the activity on its level glass surface. The reason the surface is level is because it is designed for small collisions between things like marbles and steel balls. It has to be level since the conservation of momentum in only two directions can be observed by this instrument so if it isn't level then forces such as gravity might be applied to the objects in a manner that will alter their momentum. In this case, one marble is put on a trajectory that will make it collide with another stationary marble in a glancing impact, then both marbles will be in motion and the data should prove that the momentum in both axis are conserved.








Data Collected:
Mass of marble 1 = .0864kg
Mass of marble 2 = .0857kg

The image shows the path of the marbles through the entire experiment. The blue dots are for the marble that is initially in motion and the red dots are for the marble that will be collided with marble that is initially in motion.

The image shows the data of the dots that were taken from the collision. The slopes of the data shows the velocity in both the x-axis and the y-axis for both marbles, before and after the collision. If momentum is conserved then the momentum in the x-axis and y-axis should be conserved.








Calculations:

The momentum in the x-axis before the collision = .0533
The momentum in the y-axis before the collision = 0
The momentum in the x-axis after the collision = .0747
The momentum in the y-axis after the collision = .0054

Conclusion:
Obviously the momentum in either axis isn't conserved, but that isn't proof that momentum isn't conserved. The source of error in this experiment is that the collision between the marbles wasn't a glancing collision, however that alone isn't enough to create a collision where momentum isn't conserved. The table isn't perfectly level, the marbles collided at too high of a velocity and because of the collision the rotation of the marbles might have changed. When all of these factors are combined it can be very easy for it to seem as if momentum isn't conserved.

Sincerely,

Swaggy C

Periods of Oscillation of Pendulums

Purpose:
The purpose of the experiment was to improve the ability to derive moments of inertia, and then use those moments of inertia to predict the period of oscillation as if the object were a pendulum.

Equipment Used:

The object, either a half circle or an isosceles triangle, is hung from a pivot and its period of oscillation is recorded by the photogate which records the time it takes for one oscillation to occur.











Data Collected:
First, the half sphere was oscillated, both from its base and at the apex of its curve. The only data that is needed to calculate both of the periods of oscillation for this experiment is the radius of the half circle and the location of its center of mass. The mass isn't important since the equation for torque and angular acceleration ensure that the oscillation is independent of the mass when it is acting like a pendulum.
Second, the isosceles triangle was oscillated both from the center of its base and from the tip where both equal sides come together. The only data that is needed to calculate both of the periods of oscillation for this experiment is the base and height of the triangle and the location of the center of mass.
Half Circle Data:
Radius = .125m
Center of mass = 4R/3pi
Theoretical period of oscillation from base = .77s
Experimental period of oscillation from base = .768s
Theoretical period of oscillation from apex = .755s
Experimental period of oscillation from apex = .764s

Triangle Data:
Base = .078m
Height = .145m
Center of mass = H/3
Theoretical period of oscillation from tip = .670s
Experimental period of oscillation from tip = .676s
Theoretical period of oscillation from base = .560s
Experimental period of oscillation from base = .565s

Calculations:

Half circle calculations:

These calculations are for deriving the moment of inertia of the half circle as if the axis of rotation. The moment of inertia turns out to be .5MR^2

These calculations are for deriving the theoretical period of oscillation, which is .77s. The second portion of the calculations are for deriving the moment of inertia if the axis of rotation was at the apex of the curve. It is done by using the parallel axis theorem twice, once to move the axis of rotation to the center of mass from the base of the half circle, then again to move the axis of rotation to the apex of the circle. The parallel axis theorem only works for when the axis of rotation is at the center of mass.

These calculations are for deriving the theoretical period of oscillation at the new location, which is .7548s

















Triangle calculations:

These calculations are for deriving the moment of inertia of the triangle if the axis of rotation was at the tip of the triangle. The moment of inertia was M((1/24)B^2 + (1/2)H^2).

These calculations are for the period of oscillation for the triangle if oscillated from the tip. The second portion of the calculations are for calculating the moment of inertia if the axis of rotation was from the base of the triangle. The final portion of the calculations are for calculating the theoretical period of oscillation which is .5596s














Conclusion:
For both the half circle and the triangle, the predictions were very close to experimental value, and that is because there was only a single variable to calculate for. The only reason that the prediction didn't match the actual period was because sin(angle) is approximated as being only the angle, in radians.

Sincerely,

Swaggy C

Conservation of Linear and Angular Momentum

Purpose:
The purpose of the experiment is to test the understanding of all of the concepts that are involved in this collision. The concepts are moments of inertia, gravitational potential energy, torque, and the conservation of momentum.

Equipment Used:

The device is a series of two wheels one on top of the other that has air flow through it and creates a rotating system with a negligible amount of friction. The rotating system has a sensor that reads the rotation of the wheels, both the top and the bottom, since the two wheels can be made to either spin independently or together as a single system. The hanging mass is on a pulley that has a similar design, allowing air to flow through it to create a negligible amount of friction. A small attachment is placed on the device that is going to catch a ball that will be rolled down a ramp.

Data Collected:
A ball of mass .0238kg and radius .019m was rolled down a ramp. The end of the ramp was .975m above the ground and the horizontal distance that the ball rolled away from the end of the ramp was .51m. The height up the ramp that the ball was released from was .192m. Using this data, both the experimental and theoretical velocity that the ball should have can be calculated, but that will be done in the next section. The hanging mass on the system is .0247kg and it is tied around a disk that has a radius of .05m. The average angular acceleration with the disks and the attachment for catching the ball was 5.339rad/s^2. The experimental angular velocity of the system after the ball collided with the device was 1.572rad/s

Calculations:

These calculations are for calculating the experimental horizontal velocity of the ball as it leaves the ramp and the theoretical horizontal velocity of the ball. The experimental velocity was 1.14m/s and the theoretical velocity was 1.4m/s

These calculations are for calculating the moment of inertia of the device that will catch and rotate with the ball. The device is irregularly shaped so there is not formula for deriving the moment of inertia. It was calculated experimentally by comparing the torque and the angular acceleration of the moment of inertia. The moment of inertia was calculated to be .0011191kgm^2. Once the moment of inertia is calculated then a prediction for the angular velocity of the system after the collision can be made. The predicted angular velocity of the system after was 1.712rad/s and the experimental angular velocity was 1.572rad/s.








Conclusion:
The predicted angular acceleration had a percent error of 8.91%. The most likely source of the error is that the location of where the ball collided with the device wasn't exact and that the ball didn't collide exactly in a horizontal manner. That means that the ball was either on a slightly lowered or raised trajectory.

Sincerely,

Swaggy C

Conservation of Angular Momentum

Purpose:
The purpose of the experiment is to use the understanding of angular moment, gravitational potential energy, and elastic collisions to make a prediction about the experiment.

Equipment Used:

A meter stick was pivoted as close to its end as possible and it was placed so when it would swing a piece of clay would collide with it and stick to the ruler. The piece of clay is placed as close to the bottom of the meter stick as possible and a camera is going to capture the maximum height that the meter stick will rise again.














Data Collected:
Length of the meter stick that is around the axis of rotation = .994m
Mass of the meter stick = .137kg
Mass of clay = .00955kg
Distance away from axis of rotation for clay = .994m

Calculations:

The theoretical angle that the meter stick should rise after the collision is calculated, assuming that the ruler is released from a 90 degree angle and it makes a completely inelastic collision with the clay. There is also an assumption that there is no friction and that the system didn't lose any energy when the clay and meter stick collided. The calculations predicted that the angle should have been 74 degrees and the experimental  angle was 66 degrees.











Conclusions:
The experimental angle was smaller than the theoretical angle, which isn't a credit to any error in the experiment. The source for the error was in the assumptions that the system didn't have any friction or that there was no loss of energy or momentum during the collision. The system clearly had friction and the system might have lost a little bit of energy when the clay collided with the ruler because the clay was placed on a rod and it would have taken energy to remove the clay from the rod.

Sincerely,

Swaggy C

Kinetic and Potential Energy of Magnets

Purpose:
The purpose of the lab is to develop a correlation between the function of a force and the kinetic energy of an objected that is acted on by the force.

Equipment Used:

The track is an air track which allows for a glider to move as if it is frictionless, in reality there is a tiny amount of friction but it can be considered to be negligible. There is a magnet at the end of the track and a magnet on the glider. The glider will be sent into the magnet and it will be repelled, the motion sensor at the end of the track by the magnet will be used to calculate the position and to calculate the speed of the glider before, during, and after the collision.




Data Collected:
First, a function for the force the magnet exerts on another magnet needs to be created before any experiment can be performed. This was done by raising the far end of the air track, thus having the glider have the acceleration of gsin(angle). When the glider is stopped by the magnet, then the force that the magnet is exerting at the distance away from the magnet is equal to the mass of the glider times the acceleration. As the angle became larger, the distance that separated the two magnets became smaller.

Using computer software an equation can be created for the force the magnet exerts based on the distance between the magnets. The integral of that function gives the potential energy of the magnet. The potential energy and the kinetic energy are equal to the total energy of the system, so the total energy of the entire system through the entire experiment should be a constant value. The formula for the potential energy of this magnet = (.0001344/1.011)x^-1.011.

Conclusion:
From the graph it can be seen that the red line, which is used to represent the total energy of the system, stays at a fairly constant level. The yellow line, which represents the kinetic energy of the system, is level with the total energy of the system at the beginning and at the end. The magnetic potential energy of the system is very small in the beginning and in the end of the experiment as it should be. Since the two follow the relationship that is expected between potential energy and kinetic energy then it can be said that the formula derived for the potential energy of the magnet is very accurate.

Sincerely,

Swaggy C

Work and Power

Purpose:
To develop the understanding of work and the properties of work. Then the new found concept of work will be compared and contrasted with the concept of power.

Equipment Used:

 The picture is of the stairs that the students will walk and run up. Next to the stairs over the railing there is a pulley system that has a backpack hanging over it and the students will raise it.
















Data Collected:
The data that was collected was the height of each step in a stair case and then the amount of steps in the stair case. This calculation gave the height, the delta y, that the force will be applied over.The force that is going to be applied will be the weight of the individual that is either walking or running up the stairs. The same concept is going to be applied to a pulley system with a weight at the bottom that is raised to the top. However the height of the stairs isn't going to be the distance that it will be raised will be different.

Calculations:
The force that is being applied over the height of the staircase gave work, however the time it took to accomplish that work was recorded as well. By dividing the work done by the time it took to do the work, then power is given.

Sincerely,

Swaggy C

Impulse versus Momentum

Purpose:
The purpose of this experiment was to experimentally show that impulse is equal to the change in momentum to the objected which has the force acted on it.

Equipment Used:

The cart was set with a a force sensor on top. First the cart was ran into a spring and sent back. The motion sensor recorded the position and velocity relative to time. Then a nail was attached to the force sensor and the spring was replaced and a piece of clay.









Data Collection:
For the first portion of the experiment, the cart was sent into the spring and the force sensor recorded the amount of force it took to send the cart back. The motion sensor recorded the velocity of the cart before and after the collision. The change in the momentum of the system should be equal to the integral of the force that was applied to the cart.
mass of cart + force sensor = .449 kg
integral of force = -.8135
initial velocity = 1.153m/s
final velocity = -.911m/s
Then more mass, specifically .5 kg, was added to the cart and the procedure was repeated.
mass of cart + force sensor + additional mass = .949kg
integral of force = -.8336
initial velocity = .524m/s
final velocity = -.390m/s
Then a nail was attached to the force sensor and the nail was sent into a piece of clay. The collision is considered to be completely elastic.
mass of cart + force sensor = .449kg
integral of force = -.2214
initial velocity = .760m/s
final velocity = 0m/s

Calculations:
Calculating the change in momentum is done by multiplying the change in velocity by the mass of the object. Change in momentum = m(Vf - Vo)
For the first part of the experiment the change in momentum was -.9267 and the integral of the force was -.8315. % error = 10.27%
The second portion of the experiment where more mass was added the change in momentum was -.8674 and the integral of the force was -.8336. % error = 4.05%
The final portion of the experiment where the nail stuck to the clay the change in momentum was -.3412 and the integral of force was -.2214. % error = 35%

Conclusion:
The percent error for the first portion of the experiment was fairly high. There are several reasons that could contribute to the high amount of percent error. The force sensor could have ran into the spring at an angle so the force sensor would not be able to read the entire force of the spring and only managed to read a portion of the force. The more likely reason for the error was that the cart was moving at a high velocity and it returned at a high velocity as well. This hypothesis has some validity since the second portion of the experiment, which was essentially the same except more mass was added to the cart, the cart was sent into the spring at a smaller velocity and returned at a small velocity as well. The final portion of the experiment had an extremely high percent error at 35%. The most likely source of error is that the force sensor didn't read all of the force that was being applied to the nail since the nail was taped to the force sensor.

Sincerely,

Swaggy C

Kinetic Energy of a Spring Cart

Purpose:
The purpose of the lab is to see if there is a correlation between a varying force applied over a distance and the kinetic energy of the object that is being acted on by the force.

Equipment Used:

The image is of a spring that is attached at one end to a force sensor and the other end to a cart with a block on top. At the other end of the track is a motion sensor. The force sensor will be used to measure the amount of force the spring is exerting on the cart and the motion sensor will measure the position and velocity of the cart. The point where the cart is attached to the spring's unstreched position will be the zero of the position graph. The direction towards the motion sensor is considered the positive direction.



Data Collected:

The cart was pulled towards the motion sensor and then released. The spring that was attached to the cart then exerts a force on the cart, accelerating it towards the define zero. Logger pro collected the force on the cart and position of the cart. The integral, or area under the curve, of the force graph should be equal to the amount of kinetic energy of the cart, which is half of the mass times the velocity^2.


Calculations:
When x = .193m the area under the curve was .4403 N*m and the kinetic energy is .346 J
When x = .153m the area under the curve was .5464 N*m and the kinetic energy is .433 J
When x = .110m the area under the curve was .6396 N*m and the kinetic energy is .501 J
The error at x = .193m was 21.4%
The error at x = .153m was 20.8%
The error at x = .110m was 21.7%

Conclusion:
There is a very high amount of error for every point at which data was collected, however the data is off by the same large margin for each point. This would mean that at some point during the experiment one aspect of the equipment was not properly calibrated. Perhaps the force sensor wasn't zeroed correctly or the position of the unstretched spring was not set to zero.

Sincerely,

Swaggy C

Relationship Between K, Period, and Mass

Purpose:
The purpose of the experiment is to develop a tentative relationship between the spring constant, oscillating mass, and the period of oscillation of a spring. After the tentative relationship is found then a small discussion of where this relationship comes from will be held.

Equipment Used:

The set up is of a spring and hanging from it is a mass and there is a motion sensor to record position versus time. Three other groups each had a different spring but had the same "effective" mass on the spring to oscillate.
















Data Collected and Calculations:
First, the spring constant, k, of our group's spring needed to be calculated. This was done by hanging a mass and then recording how much the spring stretched. This was repeated twice more with more mass each time this was done and then the spring constant can be calculated.
Mass = .05kg, stretch of spring = .074m
Mass = .1kg, stretch of spring = .153m
Mass = .15kg, stretch of spring = .228m
Spring Constant of our group was 6.37N/m
Second, the relationship between the period of oscillation and the spring constant was compared. Four separate groups had different springs with different spring constants but the same effective mass was hung from the springs. The period of oscillation, T, were then compared to the spring constant, k.
k = 2.39N/m, T = 1.369s
k = 6.32N/m, T = .90s
k = 14.01N/m, T = .53s
k = 26N/m, T = .42s
From the data it is seen that as the spring constant increased, then the period of oscillation became smaller. Now that the relationship between spring constant and period of oscillation has been explored, the relationship between mass and the period of oscillation will be explored.
Mass = .105kg, T = .90s
Mass = .150kg, T = 1.033s
Mass = .200kg, T = 1.125s
Mass = .250kg, T = 1.25s
From the data it is seen that as the mass increases the period of oscillation increases as well.

Conclusion:
There were no predictions in this experiment but there should be an understanding of how the relationship between the two different variable affects the period of oscillation. From previous physics problems, the period of an oscillating mass is equal to 2pi divided by omega of the system. For a spring system, omega is the square root of the spring constant divided by the mass. So T = 2pi*(m/k)^1/2. So from the equation it should be predicted that as the mass increased, then the period of oscillation should increase but if the spring constant increases then the period should be smaller and that is what occurred in this experiment.

Sincerely,

Swaggy C

Coefficients of Kinetic and Static Friction

Purpose:
The purpose of the lab is to develop a technique to find the coefficients of static and kinetic friction and then test the accuracy of the coefficients of friction.

Equipment Used:

One block with felt on the bottom of it will have a string attached to it that goes to a cup that will have water filled into using a plastic pipet. When the block with the felt starts to move then the weight of the water and the cup should be equal to the static frictional force of the felt block. The mass of the block and the cup with water will then be weighed on the scale. The same procedure will be done with the force sensor, the only difference is that the coefficient of kinetic friction will be calculated. The inclined plane with the pulley will be used to calculate the coefficient of static and kinetic friction. If the coefficient of kinetic friction is reliable then having a mass hanging off of a pulley the acceleration of the system could be calculated.

Data Collected:
mass of block 1= 108.1 ±.1g, F_pull1= .2271 ±.1N
mass of block 2+1= 249.3 ±.1g, F_pull2= .8470 ±.1N
mass of block 3+2+1= 368.2 ±.1g, F_pull3= 1.213 ±.1N
mass of block 4+3+2+1 = 495.7 ±.1g, F_pull4= 1.484 ±.1N
The angle of the inclined plane when finding the coefficient of static friction between the felt and the table top was 19, the coefficient of static friction was calculated as .3443
The angle of the inclined plane when finding the coefficient of kinetic friction between the felt and the table top was 24, the coefficient of kinetic friction was calculated as .3441, the coefficient of kinetic friction appears to be too high however it wasn't noted during the time of the experiment.

Calculations:
The coefficient of frictions for the first part were found by dividing F_pull1/Normal force

To test the coefficient of kinetic friction, a mass of known mass was held over a the edge of the inclined plane and the mass of the block will accelerate. If the coefficient of fiction was reliable then the calculated acceleration should be close to the measured value.

Conclusion:
The coefficient of kinetic friction that was determined in the last set of calculations was not very reliable. The calculated acceleration was .2618m/s² but the measured value was .5541m/s² which is a big difference. It is over a -50%, a huge margin of error.

Sincerely,

Swaggy C

Introduction to Propogation

Purpose:
The purpose of the lab is to introduce the concept of propagation, or the level of uncertainty that comes from the limitations of the precision of the equipment.

Equipment Used: 

The caliber was used to measure the diameter and the height of aluminum, copper, and steel cylinders. The scale was used to find the mass of the cylinders. After the density of the cylinders was calculated, the mass of an unknown hanging weight on two strings each at a certain tension. A tension meter was used for each string. The level angle finder was used to find the angles of both strings.

Data Collection: 

The caliber had an uncertainty of ±.01cm, so for the cylinder diameter and the cylinder height there was this amount of uncertainty. The scale used had an uncertainty of ±.1g. 

For the calculation of the unknown hanging mass one of the tension meters had an uncertainty of ±.5N and the other tension meter had an uncertainty of ±.2N. The level angle finder had an angle uncertainty of ±2 but in radians it is ±.288.

Calculations:

The aluminum cylinder had a density uncertainty of ±.1318, the copper cylinder had a density of uncertainty of ±.5816, and the steel cylinder had a density of  ±.3096.

The unknown mass had an uncertainty of ±.146kg.

Conclusion:
The uncertainty values for the density of the cylinders was very low, each one had an uncertainty of less than 2%. That is very accurate considering the units of grams and centimeters. The unknown hanging mass had a larger uncertainty value of 14.6% and that is very high.The cause for the high uncertainty came from the uncertainty in the tension meters, one of which had an uncertainty of .5N.

 Sincerely,

Swaggy C

Air Resistance

Purpose:
To develop a model for the relationship between air resistance force and terminal velocity.

Equipment Used:

Meter stick and coffee filters to be dropped

The equipment used was a 1 meter stick, 15 coffee filters, a scale, a high location to drop all of the filters and a laptop that can capture video and has logger pro.

Scale to weigh coffee filters

Data Collected:
The mass of the filters was measured, a single filter weighed .00914g.

The filters were dropped at the top of the second level and recorded as they fell. First one coffee filter was dropped then two filters were dropped stacked together and more filters were dropped until five filters were dropped together. The meter stick was used as a scale for the terminal velocity of the coffee filters. The camera of the laptop has a set frames per seconds and the velocity of the filter(s) at any moment could be determined by finding the position of the filter(s) after each frame.

Calculations:
The relationship between terminal velocity and the force of air resistance at the terminal velocity is a power law F=kvⁿ. Finding the value of n is done through plotting the terminal velocity versus the mass of the coffee filter times the constant for gravity.

According to the graph, the k value is .0052 and the power of the function, or n, is equal to 1.7751.

Conclusion:
The relationship created found during the experiment was a good model because the R²= .951 of the power trendline. All of the collected values and results weren't input into the graph to make the relationship between terminal velocity and the force of air resistance. The value of the terminal velocity for the four filters was excluded since the data point was creating a less accurate graph.

Sincerely,

Swaggy C

Elephant jet pack!

Purpose: The activity is designed to have the students look at a problem that has a non-constant acceleration and solve it using two methods, one is with calculus and the other is using excel to do hundreds of calculations to find the desired value.

Equipment: No equipment used except for this activity.

Initial problem:
The actual problem is that there is an elephant with a rocket on its back, the elephant has a mass of 5000kg and the rocket has a mass o 1500kg. The elephant is on frictionless roller skates. The elephant goes down a hill and when it reaches the flat horizontal ground it has an initial velocity of 25m/s. That is considered t=0 and at t=0 the rocket ignites and produces a constant force of 8000N in the opposite direction of the initial velocity. As the rocket burns, it burns 20kg of fuel at every second. At what position would the velocity be equal to zero?

Solution:
Method 1) Newton's Second Law of Motion states that Force=Mass*Acceleration, so there is a function for mass and a constant force so the expression for acceleration would be the force divided by the mass, which is changing.
This is completely solvable using calculus. The function of acceleration is simply the second derivative of position, so to get to the position function it requires integrating twice, also including the initial conditions for velocity before the integration of the velocity function.

The t when velocity is zero is needed. At v=0m/s, t=(19.68s) and t at v=0m/s is plugged into the position function.
Method 2) Instead of doing the tedious calculus, numerical integration can be done in Excel.

The first column is time, the second column is the acceleration, the third column is the average acceleration in the time interval, the fourth column is the change in velocity during that interval, the fifth interval is the actual speed at the specific time, the sixth column is the change in position during the time interval, and the final column is the actual position the elephant is relative to its initial point at the bottom of the hill. The first column, time, requires no calculations and is the determining factor. The smaller that the values increases by, the more accurate the other values become. The smallest value that time was set for was .05s. The second column is simply a matter of plugging in time to -400/(325-t) and finding the acceleration that that point in time. The third column is an average between the two accelerations. The fourth column calculates the change in velocity by multiply the average acceleration by the time interval. The fifth column is taking the initial velocity, which is 25m/s and adding the change in velocity calculated in the fourth column, it should be noted that it is deceleration so the change in velocity is always zero until the rocket stops and the elephant reverses direction. The sixth column is the change in position which is calculated by multiply the average velocity in the time interval by the time interval. The final column is the position of the elephant which is calculated by adding the change in position to the initial position, which is 0.

Conclusion:
The results from the Excel calculations were very accurate. The analytically calculated value for the position of the elephant when the elephant stops completely was 248.697m and the value that the numerical method resulted was 248.698m. So the error was less less than a centimeter.

Sincerely,

Swaggy C

Trajectories

Purpose:
To use the understanding of projectile motion to predict the impact point of a ball on an inclined board.

Equipment Used:

Carbon paper

 The aluminum ball is released from
a set point up the inclined portion of the ramp. It is important to have the ball released at a set point, since the horizontal velocity is going to be used to determine when the ball will impact the inclined board.

Ramp to roll aluminum ball down

Aluminum ball to roll down

Data Collected:
The data that was collected was the horizontal distance the ball traveled once the ball was sent down the ramp. The height that it fell was measured as well so the time it took for the ball to travel the distance was calculated and then the horizontal velocity of the ball was calculated.

Then the board was placed and the angle the board made with floor was recorded.

Calculations:

Conclusion:
The calculated value for the distance for the ball to impact the board was 74.7cm but the experimental value was 71cm which is very close. The percent error was only 5.21% and considering all of the variables that could affect the calculations that is very close. There could have been an error in the angle calculated for the board, or it could have shifted between measurements, the ball might have been released from a different position than initially released. Despite this, the calculation is reliable and appears to be an accurate method of prediction.

Sincerely,

Swaggy C