Lab #5
Trajectories
Ricardo Gonzalez and Josue Luna
3-15-17
The objective of this experiment is to have an understanding of projectile motion to predict the impact point of a ball (a marble in our case) on an inclined plane.
1. Introduction
We know when an object is launched at an initial velocity off of table at height (Δy), the horizontal velocity stays constant and gravity is the only force affecting an object. During the time that the object falls, the object has traveled a horizontal distance along with a vertical distance at a given time. We have set up a small experiment were we would like to know how far a marble travels at the moment the marble strikes a board of length (d) placed against the table at angle (θ). Before we measure through experimental trials what distance down the board the marble strikes, we can set up an equation using our known knowledge of kinematics to theoretically give us a distance. Having theoretically calculated the distance, we would then execute the experiment and measure the distance the marble would strike the board.
2. Procedure
We first began by gathering the needed materials and tools for the experiment, such as; Two Aluminum "V-Channels," a small marble, two wooden blocks, a string, a wooden board, a ring stand, a clamp, a white blank piece of paper, and a piece of carbon paper. We set up the apparatus in such a way that one v-channel would accelerate the marble down at an incline and then horizontally. The end of the horizontal v-channel was aligned to the same overhang length as the table and can then be used to simulate an object moving off a table with an initial velocity.
We released the marble from a repeatable point and found an area where the marble landed. On that area we first taped a blank white piece of paper on the floor and a piece of carbon paper directly above the blank piece of paper. We then released the marble from the repeatable point three times. Those three times the marble strikes the carbon paper. Interestingly, the marble created a large black dot on the white piece of paper due to the carbon paper.
Using the sting that was taped onto the edge of the table, we measured the horizontal distance the marble traveled from the edge of the table, the string, to the center of the large black dot on the white piece of paper. With the measured height and horizontal distance, we can calculate the initial velocity the marble experienced.
1. Introduction
We know when an object is launched at an initial velocity off of table at height (Δy), the horizontal velocity stays constant and gravity is the only force affecting an object. During the time that the object falls, the object has traveled a horizontal distance along with a vertical distance at a given time. We have set up a small experiment were we would like to know how far a marble travels at the moment the marble strikes a board of length (d) placed against the table at angle (θ). Before we measure through experimental trials what distance down the board the marble strikes, we can set up an equation using our known knowledge of kinematics to theoretically give us a distance. Having theoretically calculated the distance, we would then execute the experiment and measure the distance the marble would strike the board.
2. Procedure
We first began by gathering the needed materials and tools for the experiment, such as; Two Aluminum "V-Channels," a small marble, two wooden blocks, a string, a wooden board, a ring stand, a clamp, a white blank piece of paper, and a piece of carbon paper. We set up the apparatus in such a way that one v-channel would accelerate the marble down at an incline and then horizontally. The end of the horizontal v-channel was aligned to the same overhang length as the table and can then be used to simulate an object moving off a table with an initial velocity.
We released the marble from a repeatable point and found an area where the marble landed. On that area we first taped a blank white piece of paper on the floor and a piece of carbon paper directly above the blank piece of paper. We then released the marble from the repeatable point three times. Those three times the marble strikes the carbon paper. Interestingly, the marble created a large black dot on the white piece of paper due to the carbon paper.
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| This is a simplified way of visualizing our apparatus. |
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| A simple visualization of our apparatus with the board leaning on the corner of the table for a certain angle. |
Next we placed a wooden plank with one end aligned to the end of the table and the other end on the floor. We placed two large masses in front of the board in order to stabilize the board from falling down. creating a sloping angle that we measured with our phones.
Next we decided to let the marble once again roll down the v channel from the repeated point to find the area of the board where the marble would strike the board. At the point where the marble struck the board, we taped down a blank white piece of paper to the board and a piece of carbon piece of paper directly above the white paper.
The carbon paper once again helped us find the distance the marble had struck the board with the distance down the board starting at the point where the board and the edge of the table meet.
3. Measured Data
Height to the v-channel (Δy) = 0.931 +/- 0.001 m
The horizontal distance traveled by the marble to the floor (x) = 0.63 +/- 0.02 m
The angle of the wooden board (θ) = 48° +/- 1°[In respect to the floor]
4. Results
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| We first began with our basic equations of kinematics. We set Δx in terms of time and substituted the value of t into our equation for Δy. We substituted dsinθ for our Δy. |
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| We next solved for d. d in our case is the distance down the board |
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| We found the equation used to find the theoretical value of d. |
5. Analysis
We found the theoretical value of d by first calculating the initial velocity.
On the bottom of the photo above we have listed the distances for both the experimental measurement and the theoretical value. The theoretical value we have derived an equation for suggest the marble would hit the board at 0.707 meters. Also, the experimental measurement we measured through experimentation gave us a value of 0.775 meters. The difference between the two values is roughly about 0.06 meters, or, 6 centimeters. The percent error from our values is 9.6%. That error was the accumulation of the uncertainties for the values we had calculated for the height, the distance the marble struck the ground, and our measurement of the angle of the board. To expand, our equation for the theoretical value of d is dependent on our initial velocity and angle, both of which have a range of error.
6. Conclusion
When we calculated the theoretical value for d, we did not take into account of any other forces onto the marble during its flight time other than gravity. Although air resistance is present, given the small surface area of the marble, we completely ignored the air resistance force that would not have affected our theoretical value. When measuring the experimental value, we used a meter stick to measure the distance from the top of the board to the black dot created below the carbon paper on the blank white piece of paper, which was not perfect. A range of uncertainty should be accounted for. Technology has advanced so much in recent years, one of which is the ability to measure angles with our phones. The measurement we took was not accurate to the nearest whole number and thus our value of uncertainty of one degree was relatively large in comparison to our measurements, in respect to a percentage of error.
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