Projectile Motion Lab (Kinematics)
Summary of Work
In this project, we are trying to figure out if the equation, x=vx0(2yay), Will be able to accurately predict the horizontal distance a projectile will travel after being dropped from a specified height on our ramp, and the ramps height above the floor. Our hypothesis was We can predict the horizontal distance a projectile will travel based on exit velocity and vertical displacement within an 8 percent margin.
We first had to heighten our understanding of kinematics, we did this through studying chapter two in the college physics book. This allows us to create graphs, accurate data measurements, and create equations using the data we are given. We derived the time equation for projectile motion to allow us to calculate the velocity without a time recording, which was essential because we couldn't accurately record the time. Using this newfound knowledge, we created an equation that could predict the horizontal distance a projectile would travel when starting from a predetermined height. We also had to assume that the acceleration due to gravity was 9.8m/s^2, and that the velocity and acceleration equations were true.
After performing a test using all the concepts we practiced, we came to the following conclusion: Using exit velocity, vertical displacement, and average landing position from a consistent exit velocity, you can predict the horizontal displacement of a projectile (within a margin of error) that is launched on the same ramp used to gather the initial data, placed on varying heights. To start this experiment, we needed the data listed above. To start, we constructed a ramp (See blueprint materials needed section). Using this ramp design, we record horizontal displacement of our projectile from varying starting heights on the ramp. Other measurements that were essential were the vertical displacement and vertical acceleration (while free falling). It is important to note that we only required data from AFTER the projectile left the ramp. This is because the equation we were using to predict horizontal displacement only included projectile measurements. To start, we recorded that when starting our projectile from 0.42m up on our ramp, it would travel an average of 1.032m horizontally before hitting the ground. To calculate the exit velocity and time which are essential for the “prediction” equation, we needed to first find the time in air. We could do this using the equation t=(2yay). We knew that the change of y (vertical displacement) was 0.855m. We also knew that the acceleration of y (acceleration due to gravity) was 9.8 ms/^2. The time in air was then calculated to be 0.417 seconds. Now with the time, and the already determined change of x, we can calculate the velocity using Vx=xt. The exit velocity was 2.48m/s. Now we have enough information about our projectile being dropped from 0.42m up on our ramp to create an accurate prediction using x=vx0(2yay). The perch that the ramp will be placed on is 0.855m above the ground. That will be filled in for a change of y. Our acceleration due to gravity is -9.8 m/s.^2, that will be used for Ay. Exit velocity is 2.48m/s. That will be used for Vx0. Now all that's left to do is to solve! The equation predicts that our projectile, starting from 0.42 meters on our amp, and placed 0.855 meters above the ground, will travel 1.116 meters until making contact with the floor. When testing this in real life, we created a margin of error. We did this as there are factors we cannot control, like friction, a symmetrical ramp construction, and small measurement issues. We did use an iphone camera and meter sticks after all. Our margin of error was +/- 2.8% of the predicted impact spot (1.116m). We ended up landing within that margin every single time.
Physics Concepts
Time in air
t=(2yay)
t=(2(-0.855m)-9.8m/s²)= 0.417s
We used this derived equation because it consisted of all the data we already had figured out/knew. We used our tools to measure vertical and horizontal displacement. We also knew the acceleration due to gravity because, well, we tested this on earth.
Exit Velocity (Horizontal)
Vx=xt
To find our velocity, we first needed the average change of X. We luckily measured the horizontal displacement of our projectile three separate times for each height. This allowed us to create an average. Using these average changes of X and the time in air (calculated above), we were able to find the velocity of all 4 trials.
Horizontal distance with varying vertical displacement
x=vx0(2yay).
Mr. Williams helped us derive the equation above from one of the pre-existing equations on our AP Physics Everything sheet. The equation calculates the change in X when you plug in different vertical displacements. This was key for the second part of our project where Mr. Williams would give us a random vertical height to place our ramp on and then try to estimate the landing position. The starting height we received and had to calculate for was 1.166 m. Here is the work:
x=2.55m/s(2(-1.166m)-9.8m/s²)
x=2.55m/s(0.487s)
x=1.242m
Margin of error
Due to friction, measurement errors and reconstruction, we had to create a margin of confidence. What this means is that we had to determine an area above and below the estimated landing position of 1.242 m. We choose a margin of error of +/- about 8% (The piece of binder paper). Mr. Williams was much more optimistic than us and gave us a separate margin of error (the pink post it note) had about a +/- 2.8% degree of confidence. We ended up hitting that note every single time! (proved by the paint marks our car left on impact)
Procedure
Brainstorm and plan a ramp design that can be used from the top of a table to send a projectile off of it in a consistent manner.
Construct your blueprint. (Make sure that it can be consistently replicated). Include two meter sticks going away from your ramp horizontally so you can measure your projectiles distance.
Mark out 4 separate heights on the ramp to start your trials from, make sure that they are reasonable heights that can be used to create accurate equations.
Release your car 3 times from each height and use a slow motion camera to record it's horizontal displacement.
Once you have these measurements, use the derived equation for time (t=(2yay)) to find it's time in the air.
Now, with accurate time measurements, calculate the exit velocity of your projectile (The velocity when it leaves the ramp). Use this equation: Vx=xt
Now, you should graph your position vs. time, velocity vs. time, and acceleration vs. time using your new measurements.
Use the equation x=vx0(2yay) to calculate your projectile's landing position at any given exit height (The height above the ground where you're projectile LEAVES the ramp). Plug in the height of Mr. Williams desk along with your other data points. You should now have an estimated landing position.
Using your own wits, create margin of error for your projectile. (an area where you estimate your projectile to land.)
Now, recreate your ramp on Mr. Williams desk exactly as it was before to maximize your accuracy
Launch your projectile at the height you used in your equation from your ramp. If you did everything correctly, your projectile should consistently land in your margin of error.
Data
Reflection
During my first semester of AP Physics, I learned various things about myself. Firstly, I learned that I was a lot more capable than I thought. Most of my struggles in this class have just sprouted from not being confident in myself. I would always predict myself failing the tests, but at the end of the day i was getting a consistent flow of 3's and 4's. I learned that I work best in environments where I have a set goal. I can fill in the blanks to get to the finish line easily, but when I cant see the end, it makes it very hard to do everything in the middle. I think its important to know where you do your best work. Reflections like these and the experiences I have in such a mature environments like AP Physics are doing more than just heightening my knowledge of Physics, they are setting me up to live my best life.
As far as things I need to work on goes, well lets just say the list is never ending. Things that I believe I can change in this class are the following. To start I tend to get lazy when I feel overwhelmed, and when you are lazy and don't do your work, things just get more and more overwhelming. Its like exponential growth. I can certainly focus and pushing through the beginning of a project so I can stay ahead of my workload. This is just another internal conflict that is setting me up for failure. Another thing I can work on improving is my teamwork. I believe that I have a lot of really good ideas (Key word: believe). I try to force my ideas upon my classmates and get disappointed when they are rejected. I think its because I put a lot of effort into my ideas, and think them through all the way till the end. I know that they will work but I cant explain it, so its obvious why some of them are rejected. I can fix this by either becoming better at explaining my ideas, or be more open to new ones.
Six C's
I can say with confidence that critical thinking is one of my strong suits. Or at least I thought it was before taking this class. I use this skill everyday in class, and I can tell its becoming stronger. I was used to using critical thinking for more creative applications, rather than logical, numerical uses. Conscientious learning is something that I think I can work on. I try to pay attention as much as I can but it can be hard with such complicated, equations. I think deriving equations is where I fall behind and normally stop paying attention. Hands on experience is something that I'm sure can help me, I just need the fundamentals to start. Collaboration and Communication are both my strong suits and downfalls. As I explained above I am what some people might call stubborn. On the bright side, I believe that my directness and honesty can be very helpful. While I recognize I come of as harsh sometimes, I think that being real and not trying to sugarcoat anything is always better than lying. I know that personally I would always want someone to tell em if I'm wrong or spark debate with me rather then agreeing with everything I say. I guess it might be presumptions of me to think that people want the same thing, so I hope people will tell me if my confrontations is not what they want in a team.