Projectile Lab - MarcusL and IsabellaW

Abstract: Utilizing a launcher, the task is to find the initial velocity of a marble shot out at a cannon. Based on that result, and the variance of initial velocity, the variance of the landing position of the marble can be determined in an effort to hit a target of a specific diameter. Through analyzing 4 videos of marbles shot out of the cannon in logger pro, an average initial marble velocity of 3.9906 ( m / s ) was determined. In an attempt to calculate the variance of the X-landing position, we factored in the uncertainty of initial launch velocity and angle, and it was ultimately determined to be ± 0.076 m . Ultimately, our cannon setup hit the large aluminum plate target (⌀22.5 cm), but missed the duct tape target (⌀8.9 cm) as expected from our variance. Procedure: [Set up] 1. Set the cannon to the minimum height on the slider, so that the base of the cannon is in contact, and is parallel, to the desk-surface. This means that the cannon should be vertical – check that the dangling weight and string shows that the cannon is at 90˚ 2. Put the marble into the cannon, and push slowly until you hear two clicks (we are using the second power level). 3. Place a meter stick parallel to the face of the cannon (the face with the labels) and align it horizontally with the center of the marble. Picture of our launch setup.

[Data recording] 4. Start recording a video (HD, 60 FPS), make sure the camera is at least three meter away from the cannon to avoid parallax from the camera. 5. Pull to release the cannon. 6. When the marble falls back down and passes the “release point” of the cannon (the label on the cannon which marks where the ball is no longer in contact with the cannon), stop recording. 7. Upload the video to logger pro for analysis. 8. Use the “dimension tool” to add proportion to the video: drag a line from the bottom of the meter stick to the top, and dimension that as 1 meter. 9. Skip ahead in the video to the first frame in which the launched marble is first visible. 10. Use the “plotting tool” to plot the first point. Then, the video should cut to the next frame. Plot a point where the marble is again. 11. Repeat step 11 until the ball is at its maximum height. 12. Plot five more points after its maximum to ensure that we have captured the maximum height in our data. 13. Next, use the “measure tool” to measure the distance between the “release point” of the cannon and the maximum. Record this number (meters). [Finding initial velocity] 14. Use the kinematic equation, V y 2 = V 0 y 2 + 2 a Δ y to find initial velocity of the cannon, V 0 y (see calculation [1] below). 15. Repeat steps 3-14 three times. 16. At this point, there should be four calculations of initial velocity. Subtract the lowest calculation from the highest to find the Range of the velocities for later variance calculations. 17. Find the average of the four calculations to use as the initial velocity for the shooting test calculation (see calculation [2] below). [Data usage reasonings & calculations] 18. The formula △ y = tan ( α 0 )∗ x − 1 2 ∗ g V 0 2 ∗ co s 2 ( α 0 ) ∗ x 2 plots the projectile motion of our marble in an attempt to hit our target. To determine the horizontal X-position of the cannon relative to our target from this formula, we set an angle of the cannon and the Y- position of the cannon. 19. Choose an initial vertical launching position for the cannon. To measure this vertical position, use a meter stick to measure the distance between the “release point” and the ground. Ensure that the meter stick is perpendicular to the ground. Record this measurement.

3. Use a ruler to take the vertical elevation of the cannon for △ y : -76.7cm = - 0.767 m. 4. Calculating the desired x distance between the launching point and the target from Step 24 : △ y = tan ( α 0 )∗ x − 1 2 ∗ g V 0 2 ∗ co s 2 ( α 0 ) ∗ x 2 − 0.767 = tan ( 45 ˚ )∗ x − 1 2 ∗ 9.8 3.990619547 2 ∗ cos 2 ( 45 ˚ ) ∗ x 2 − 0.767 = 1 ∗ x − 4.9 3.990619547 2 ∗ cos 2 ( 45 ˚ ) ∗ x 2 Solve the quadratic for x x = 2.193 m ( 3 s .f . ) , reject the negative solution, since we are launching in the positive direction. Calculations 1-4 provide us with a good, but arbitrary estimate of the x- displacement that we should expect from the launch. We can push our calculations further by taking into account variance. 5. There are two sources of uncertainty in our experiment: the initial velocity and the angle of launching (specifically because our string to measure angle was crooked). We need to take in consideration of both to find the variance of our x-landing position. a) To calculate the upper-bound x-displacement, we can try a smaller launch angle of 44.5˚ (assuming that the lower angle will increase range) and maximum initial velocity that we found in Procedure Step 16. Upper-bound x-displacement: V 0 y = 4.008256479 m / s △ y =− 0.767 m,α 0 = 44.5 ˚ △ y = tan ( α 0 )∗ x − 1 2 ∗ g V 0 2 ∗ co s 2 ( α 0 ) ∗ x 2 x = 2.216 m ( 3 s.f . ) b) To calculate the lower-bound x-displacement, we can try a larger launch angle of 45.5˚ (assuming that the larger angle will decrease range) and the minimum initial launch velocity we found in Procedure Step 16. Lower-bound x-displacement:

V 0 y = 3.938213696 m / s △ y =− 0.767 m,α 0 = 45.5 ˚ △ y = tan ( α 0 )∗ x − 1 2 ∗ g V 0 2 ∗ co s 2 ( α 0 ) ∗ x 2 x = 2.140 m ( 3 s .f . ) c) Finally, using the upper and lower bound x-displacmeents, our final variance can be calculated. Range of Variance: 2.216 m − 2.140 m = 0.076 m = 7.6 cm Average x-displacement: ( 2.216 m + 2.140 m )/ 2 = 2.178 △ x = 2.178 m,± 0.076 m 6. Deciding the target: With our calculated variance, the total x-range in which the projectile could land in is 7.6 cm ∗ 2 = 15.2 cm . With this maximum range, the plate will be a sure hit since 15.2 cm fits comfortably within the diameter of the plate. Results: Using 2.178 m as the horizontal distance of the cannon away from the metal plate, we launched the cannon. It hit the rim of the plate and hit as expected according to our variance calculations. Here is our recorded video of our first shot (I hope you enjoy our elation as it hits the target): https://drive.google.com/file/d/1tn0BDTfG-KFK7Lg-gu-5JBdKCBIq9JmJ/view?usp=sharing Afterwards, we tried to hit the roll of duct tape. According to our variance calculations, the marble may miss since our variance is greater than the diameter of the roll. We missed the tape on our launch. Here is the video of our second shot: https://drive.google.com/file/d/1_iZ8ALS9b4t0fMsMLr2q_oRy5_GVysmq/view?usp=sharing Error Analysis: This section will briefly discuss the sources of variance, and how these sources can be minimized if we were to repeat this experiment. 1. The error may be negligible here, but our launch video analyses may be slightly inaccurate due to the camera’s perspective. We didn’t stand far away enough from the cannon and film at an elevated position to reduce parallax error when measuring distances. Next time, we would have the camera horizontally aligned with the middle of the meter stick, and film at least 5 meters away from the cannon.