Table of Contents:

Final Project

Roller Coaster Project

Final Project

Evidence of Work:

The purpose of my final project was:

• To predict and compare the order that a ping pong ball, golf ball, basketball, and beach ball would hit the ground when dropped from a height of 5 feet and 15 feet.

• To calculate the velocity, acceleration, Fg and drag Force on each ball at 5 feet vs 15 feet

• To predict whether any ball will reach terminal velocity and if so, compare calculated terminal velocity to measured terminal velocity

• Further, I intended to create an educational presentation on my findings and the physics of a free fall (see attached).

To accomplish this, I used a tape measurer to measure 5-foot distance from my extended arm to the ground, and a 15-foot distance from a balcony to the ground. With an i phone video camera, I had someone take a video of me dropping each type of ball from each height. I repeated each ball drop twice. I analyzed the videos to count the number of frames and calculated the time of each drop. I also weighed each ball and measured the diameter of each ball and compared these values to measured standards.

I used the equation Velocity = change distance/change time to calculate the velocity for each ball drop and then used these velocities to calculate the acceleration for each drop with the equation Accleration = change velocity / change time.

I used Newton's second law with acceleration of gravity (F=mg) to determine the Force of gravity on each ball. I used the drag equation to determine the effect of air resistance on each ball. I compared the values of Force of Gravity and Drag Force to determine if they were close to balance for any of the balls, and then used the terminal velocity equation to test whether these balls had reached theoretical terminal velocity.

I found that at 5 feet the balls all hit the ground at a similar time, although the beach ball was slightly slower at 15 feet the golf ball hit the ground first, follwed by the basketball, ping pong ball and the the beach ball. The beach ball had balanced Force of Gravity and Drag Force and reached terminal velocity. I described and analyzed my findings in a narrated presentation. The presentation is formatted like a scientiffic write up, except that I included educational descriptions of the physics of a free fall in both the Introduction and Discussion sections.

Project Progress Log

Content

Newton's second law says that when a constant force acts on a massive body, it causes it to accelerate, i.e., to change its velocity, at a constant rate. For a free fall F=ma becomes F=mg where g = gravitational acceleration or 9.8 m/sec ^2.

In a vacuum, all objects free fall with acceleration equal to gravity. They will get faster and faster until they eventually hit the ground. In air, however, there is a larger force of gravity on more massive objects. There is also air resistance. In physics, air resistance can be calculated as Drag Force. The force of gravity causes the ball to move towards the ground and the Drag force (due to air resistance) slows down the fall. These forces act opposite of each other.

Drag Force can be calculated using the drag equation. Drag Force=the Drag coefficient x the density of air x velocity squared x the projected area of the ball divided by 2.

The Drag coefficient is based on the shape of the object, in this case for a sphere, it is 0.5.

Projected area is the part of the object that is going to encounter the air resistance. It’s kind of similar to the shadow that the object would make. For a sphere, it is easy! Projected area is just the area of a circle with the same diameter as the sphere.

If you look at the Drag Force equation you can get an idea of the factors that increase Drag or air resistance. If velocity is higher it should increase air resistance. This is easy to imagine if you think about sticking your hand out a car window. As the car speeds up, there is more air resistance against your hand (basically because your hand is hitting more air faster).

The other numerator in the drag equation is Projected Area. Increasing the projected area of an object will also increase Drag. A parachute opening is an easy comparison here. As the projected area of the falling object increases, the acceleration slows.

Terminal velocity is another concept important to a free fall. In a vacuum, there is no terminal velocity. Objects all free fall with acceleration equal to gravity. Remember, they will faster and faster until they hit the ground. In air, terminal velocity is different for different objects based on their mass, the drag coefficient of their shape, and their projected area

The Formula for terminal velocity is the square root of 2 times mass x acceleration of gravity divided by the density of air x projected area x the drag coefficient for the shape. When Drag force = mg, so that no more acceleration is occurring, then terminal velocity is reached. So if we set the drag equation as equal to mg we can see how to derive the equation

This means that, in air, objects accelerate more quickly at first and then acceleration slows as air resistance increases until terminal velocity is reached. It also means, the higher air resistance is, the lower terminal velocity will be

The ball drop times and velocities are shown in the table below

The calculated accelerations are shown in the table below

These values are exprssed in the following line graphs:

Between 5 feet and 15 feet, the velocities continue to increase. The acceleration continues to increase for the basketball and golf ball, but not as quickly…. and it starts to decrease for the ping pong ball and especially the beach ball

Force of Gravity for each ball calculated by F=mg

Ping pong ball: 0.0265 N

Golf ball: 0.4478 N

Basketball: 6.1152 N

Beach ball: 0.098 N

I calculated drag force for each ball at 5 feet and 15 feet shown in the table below. Drag force increased between 5 and 15 feet for all the balls.

I then made bar graphs to compare the forces of gravity and drag forces between the balls. I then compared the force of gravity and drag force for each ball which showed that the forced were essentially balanced for the beach ball indicating it has reached terminal velocity

At 15 feet (at which point acceleration is decreasing due to increased air resistance):

• • • The golf ball hit the ground first due to its greater acceleration compared to the other balls. This is because it had enough mass to have a significant force of gravity, and little air resistance due to its small size. The height of the fall was also not great enough to have air resistance slow it down much.

    • The basketball was second because of its greater mass (force of gravity), and inadequate height of the fall for air resistance to increase enough to slow it down. It had more air resistance than the golf ball due to its size

    • The ping pong ball was third because it had low mass and low air resistance

    • The beach ball was the last to hit the ground because of its low mass (less force of gravity) and larger diameter (increased air resistance).

• At 5 feet, (when the force of gravity predominates):

  • • The balls all hit the ground at a similar time, although the beach ball was slightly slower as it encountered some air resistance even at the beginning of it’s fall.

    • The beach ball (which had the lowest velocity) was the only ball to reach terminal velocity during the 15-foot drop.

    • The measured Final velocity was almost identical to the calculated terminal velocity

Two things I did well, in this project were conscientious learning and communication. I demonstrated conscientious leaning because I was careful to manage my time in doing this project and creating the presentation. My health has been very unpredictable lately and I wanted to make sure I had time to complete it. It was lucky because I got quite sick during finals week and if I had procrastinated, I never would have had time to complete the project and create this website. I improved upon my communication skills. I spent a lot of time making sure that my presentation slides were clear and thorough so that someone who did not know the topic could learn the material from my educational presentation.

This project was not a good example of collaboration. I knew I might not be able to collaborate on a project because of my illness, so I designed this to be an independent project. But because of this, it did not help me work on my collaboration skills. Even though doing this on my own gave me a chance to take responsibility for all aspects of the project, I have learned that I really do prefer working in groups. I will try to look for more opportunities for collaboration in the future. Also, even though I though I did a good job with time management in getting the project and presentation done early, I waited to do the website update, which I regretted after I got sick. I also didn't try to upload my video to google classroom until the night before, not realizing it would take literally forever, due to the very large file size. Lesson learned.

Roller Coaster Project

Evidence of Work:

For the Roller Coaster project, our problem was to try to measure all types of energy within a roller coaster model we designed and built, including spring potential energy, gravitational potential energy, kinetic energy, and thermal energy. We attempted to track energy transfers from one type to another at different points along the track and we hoped to demonstrate that energy was conserved within the roller coaster system.

My group designed a track with a spring starter, an elevated starting point followed by an incline, a decline, and straightaway and a loop (pictured below).

After building the track we measured the lenthgs along the different segments of the track, and we measured the mass of the car we were sending on the track. The mass of the car was 0.028 Kg. A diagram of the lenths of the track is shown below:

We identified 6 points along the track to determine the amounts of different energy and energy transfers:

1. Starter

2. Right after starter

3. Peak of track

4. Right before loop

5. Top of loop

6. End of track

We measured the height of the track relative to the ground at each of these points (see below):

We sent the car on the track 3 times and took a video with a timer in the video backgound. By analyzing the videos, we were then able to calculate the mean time it took the car to get to each point on the track and then calculate the velocities using the equation velocity = Change distance / change time.

We then set out to calculate the various energies at each point along the track as outlined in the content section.

We then created a google slides presentaion (attached below), and presented our results to our classmates. Our presentation outlined our approach to calculating the different energies at each point along the tract based on the pricipal of conservation of energy. We showed that the relatively equal amounts of gravitational potential energy and spring potential energy just at the start were converted to other types of energy including kinetic energy and thermal energy as the car rode the track. However we did not manage to show conservation of energy. We calculated some energy loss, likely due to a combination of measurment error and likely some mathematical errors.

Content

Spring potential energy is the energy, stored in a compressible or stretchable object like a spring. It can be calculated by using the equation Usp= 1⁄2 K X^2. where k=Spring constant and X = displacement. The spring constant is different for different springs. The combination of how far the spring is displaced and K determine the spring potential energy. We calculated the spring energy below:

Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. We used the mass of the car and height of the track at different points to calculate the Enegy of Gravity at the starter using the equation Ug = mgy (Ug = Potential gravitational energy, m=mass, g=gravity, y=height)

Then since at the starter all energy should be either gravitational or spring potential, the total energy at the Start was calculated as: Gravitational Potential energy + Starter Spring energy or (Mgy) + 1⁄2 K X^2 = Total energy

To determine the energies right after the starter we had to consider kinetic and thermal energy as well. When the the spring starter fired, the spring potential energy should be converted to kinetic energy. Kinetic energy was be calculated using the equation: KE=1⁄2 mv^2

A smaller amount of potential energy would be converted to thermal energy due to friction.

The equation for thermal energy is: Eth or Work = Friction x Displacment or W=F x Change D

Thermal energy could not be directly measured because we did not know the friction, so we used the equation Ugi + Ki = Ugf + Kf + Eth to solve for thermal energy which is equivalent to: mgyi + 1⁄2 mvi^2 =mgyf + 1⁄2 mvf^2 + Ff (change X)

Gravitational potential energy should be unchanged since the track height is unchanged right after the starter.

As the track rises to the peak of the track gravitational potential energy will increase dur to the increasing height and kinetic energy will decrease. The same equation can be used as seen below:

In order to find the kinetic energy right after the superslide we used the equation:

Ugi=Kf +Eth Ugf

mgyi=1⁄2 mv^2 + Ff (change X) + mgyf

In retrospect, we should have included initial kinetic energy and Ugi on the left hand side of the equation as the car had kinetic energy going into the slide. This equation would have been correct if the car had not been in motion prior to the slide That is likely one of the reasons we failed to show conservation of energy at every step.

We did see gravitational potential energy being converted to kinetic energy and thermal energy as seen below:

In order to find the kinetic energy and thermal energy for the loop de loop we used a modified equation to account for the increased friction in the half pipe

Ugi+Kf +Eth = Ugf+Kf +Eth+Eth-pipe

mgyi +1⁄2 mvi^2 + Ff (change X)=mgyf +1⁄2 mv^2 + Ff (change X)+Eth-pipe

However, in retrospect we made an error in how we calculated velocity in the loop. We used the minimum velocity equation to calculate the velocity of the top of the loop (sqrt of g x r). The minimum velocity is the minimum velocity required to keep the car from falling to the ground at the peak of the loop. This likely under measured velocity in the loop slightly.

The following graph shows the comparison of energies at each point on the track. we did show energy converting from spring potential energy and gravitaional potential energy at the beginning of the track, to mostly thermal energy by the end. We did not show conservation of energy. Energy went unnaccounted for at multiple steps. Some of this was likely due to measurement error. However, some was likely due to some reasoning errors like not including kinetic energy as an input at the beginning of the superslide.

Two things I did well in this slide were working with my group mates and communication. This was a fun group project to do and I think all of us in my group worked well together in completeing the experiment. One of my group mates took the majority of our notes in class, so I took the lead in creating the presentation. This cooperation allowed us to have a successful project. I also think I did a good job with communication. I tried to create a detailed presentation that communicated our work and thought processes. Writing out a presentation in a systematic way is a good way to communicate findings, but I have found it also helps me organize my thinking about a topic.

Two things I could have improved on this project were critical thinking and time management. We made some errors in reasoning out equations for this project, probably because we were too eager to use a formula we were familiar with that seemed to fit. For example, not inluding kinetic energy at the beginning of the superslide, because we were familiar with a scenario of an object sliding down a slope where graviatational potential energy was the only starting energy. I would improve upon this in the future by critically thinking about each step before rushing to use a formula. Second, I didn't start working on the actual presentation for this project until a couple days before it was due. I think if I had had more time, I could have made teh presentation even stronger and maybe caught some of our errors ahead of time. I have found it takes quite a while to put slides together, so I would definitely give myself more time in the future, like I did with my final project.