Figure 1: The trajectory of a projectile launched at θ 1 > 0.

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1 3 Projectile Motion Introduction Important: Complete and submit the Lab 3 problem set on WebAssin before you leave lab today. Your instructor and lab assistant will be happy to help. In Lab 2, you tested a model for position and velocity as functions of time for objects movin in one dimension with constant acceleration. You will now have an opportunity to test the same model in two dimensions throuh the study of the motion of a small spherical projectile fired from a sprin-loaded launcher. You will be able to measure initial velocities (both manitude and direction) as well as vertical and horizontal displacements and compare your measurements with model predictions. Once you have had a chance to study the system, you will be challened to land a projectile on a taret placed by your instructor. v θ y x Fiure : The trajectory of a projectile launched at θ > 0. Our model for the velocity of an object movin in two dimensions under constant acceleration is v 2x = v x +a x t () v 2y = v y +a y t (2)

2 The horizontal and vertical components of the displacement of the object as functions of time are iven by x = v x t+ 2 a x t 2 (3) y = v y t+ 2 a y t 2 (4) where x = x 2 x and y = y 2 y. The trajectory of a projectile in free fall under the influence of ravity is illustrated in Fi.. In this case, we have y 2 = 0, v x = v cosθ, v y = v sinθ, a x = 0, and a y =, ivin and v 2x = v cosθ (5) v 2y = v sinθ t (6) x = (v cosθ ) t (7) y = (v sinθ ) t 2 t2 (8) where is the local acceleration due to ravity. Youwillshowforhomework, bycombinineq. s7and8,thatthehorizontalraner = x of a projectile launched with initial speed v, at an anle θ with respect to horizontal, from a heiht y and landin at heiht y 2 = 0 is iven by R(θ ) = v cosθ v sinθ + ( v sinθ ) 2 + 2y (9) This is the key equation for findin a launch anle that will ive the projectile a particular horizontal rane. Unfortunately, Eq. 9 cannot be solved for θ alebraically. You will use a spreadsheet to solve it numerically instead. Experiments You have been supplied with a projectile launcher, a plastic ball (your projectile), a meter stick, and a sheet of carbon paper. Accordin to the manufacturer of the launcher, the launch position (x, y ), the position at which the ball loses contact with the sprin and enters a state of free-fall, does not vary with launch anle. However, the initial speed imparted to projectiles may vary as much as 8% over the available rane of launch anles. Use the medium rane settin of the launcher for all measurements. Preliminary Launches. Measure y, the heiht above the table at which the projectile is launched. Be careful here. You want the manitude of the actual vertical displacement of the ball.

3 2. Launch the projectile at θ = 0 with respect to the horizontal. (See Fi. 2.) It should land on the table before bouncin away. If it doesn t, mount the launcher further from the end of the table and try aain. 3. Now you know approximately where the ball hits the table for a horizontal launch. Tape a blank sheet of paper over this landin zone, and lay a sheet of carbon paper over it. If you have placed the paper properly the projectile will touch down on the carbon, leavin a mark on the paper. You can then use a meter stick to determine the horizontal rane x of the projectile. 4. Launch the projectile several times to produce a hit pattern which ives you a sense of the reproducibility of the horizontal rane. 5. Then, set the launch anle at θ = 80, and measure the maximum rane for several launches. The point of these two initial experiments is to ive you a chance to test both the launcher and the theoretical model so that you can make an informed prediction for the Taret Practice experiment. Skip to the Analysis section before proceedin. Taret Practice Now that you have studied the launcher and your theoretical model, see if you can use what you have learned to hit a taret. Your instructor will tape a taret to the lab table. Landin the projectile within the boundaries of the taret on the first try is worth extra credit. Have your instructor witness your attempt. You may use only the meter stick and a spreadsheet to choose your launch anle. Trial shots (beyond your preliminary launches at θ = 0 and 80 ) disqualify you for extra credit and invite a rade penalty. Analysis Initial Speed from a Horizontal Launch The trajectory of a projectile launched at θ = 0 is shown in Fi. 2. You can determine v from your measurements of the horizontal and vertical displacements of the projectile y and x. The time it takes the projectile to drop to the table is found usin Equation 8. The projectile was launched horizontally, so v y = 0. Hence, time at which y = 0 (i.e. when it hits the table) is iven by 2y t max =. (0) The initial velocity is then iven by pluin t max into Equation 7, v = v x = x t max. ()

4 v x (v = 0) y y x Fiure 2: The trajectory of a projectile launched at θ = 0. Launch at an Anle In order to determine the launch anle you will use to try to hit the taret, proram Eq. 9 in a spreadsheet so that you can adjust the parameters easily. Compare the predictions of this theoretical model with your measured horizontal ranes with θ = 80. How well do they compare? Is there any pattern in deviations of the model from experiment? Do you need to correct the model in order to hit a taret? If you aren t familiar with spreadsheets or need a refresher, here is one way to proram Eq. 9. Leave row A for column labels. Put the initial heiht in cell B, the initial speed in cell B2, the launch anle in cell B3, and in cell B4. Place v sinθ in cell B5, which in spreadsheet syntax looks like =B2*sin(radians(B3))/B4 Finally, place the rane in cell B6, =B2*cos(radians(B3))*(B5 + sqrt(b5^2 + 2*B/B4)) The spreadsheet tri functions assume anles are expressed in radians. I use the radians() function above to convert to radians so that the launch anle in cell B3 can be entered in derees. Oranizin the calculation in one row facilitates producin several rows with different parameters (y, v, θ,, e..) for comparison. Uncertainty Your distance and anle measurements have inherent uncertainties. Estimate and record these. Use the hit patterns to quantify the uncertainty in horizontal rane. Takin Your Shot Before you attempt to hit the taret, o over your calculations with your instructor/ta and be sure to have your instructor/ta present to witness your shot.

5 Questions As you formulate answers to the followin questions, apply the concepts related to systematic uncertainties and uncertainties due to random variations (Appendix A) that you have been learnin in lab this semester.. Did your launches at θ = 80, lead you conclude that you needed to make some kind of correction to the model in order to hit a taret? Explain. If you made a correction, describe it. 2. Do the uncertainties in your meter stick and plumb bob measurements explain the random scatter in the observed hit patterns? Explain. If not, what else miht be oin on? 3. Based on your work, what minimum taret size can you reliably hit with your launcher? 4. If you hit the taret, was there any luck involved? If you missed the taret, why do you think you missed? Did you make a mistake or were you a victim of random variation?