Enerizin Math with Enineerin Applications Understandin the Math behind Launchin a Straw-Rocket throuh the use of Simulations. Activity created by Ira Rosenthal (rosenthi@palmbeachstate.edu) as part of the NSF-funded InnovATE rant. Presented at the Back to School Math Conference Santaluces Hih School Auust 8, 016
An application of Projectile Motion: Launchin a Rocket (or a Baseball or a Golf Ball etc ) Learnin Outcomes: Upon completin this activity, students will be able to: Understand the relationship between a mathematical formula and what it represents in real life. Investiate relationships between two variables throuh the creation of a scatter plot and throuh the analysis of the equation that relates the variables. Solve a iven formula for one of the variables, in order to make a projectile hit a preset taret. Terminoloy: The distance traveled horizontally from the launch position to the landin position is known as the rane, shown as R in the iven fiure. The maximum heiht reached by the projectile is shown as H, and the launch anle is marked as θ. The initial velocity of the projectile is represented as Vo. For all of the followin questions, we will assume that the initial heiht is 0, that is the object is bein thrown from the round level. Use a rocket launcher or a simulation website such as the one found in https://phet.colorado.edu/sims/projectile-motion/projectile-motion_en.html to answer the followin questions. 1) Initial Guess: How do you think the rane of the projectile will chane as the anle chanes (assumin a fixed initial velocity)? How do you think the rane will chane as the initial velocity chanes (assumin a fixed anle)? How do you think the maximum vertical distance from the round will chane as the anle chanes (assumin a fixed initial velocity)? 1
) Experimentation and Data Collection: Please note that you won t be able to use the values iven at the top of the simulator, since the projectile in that simulation is not thrown from (0,0) level. Instead use the tape measure to determine the rane and the maximum heiht. Measure from and to the ray horizontal line, as this represents the level from which the object is launched. For the rane, the tape measure should be set so the plus sin is alined with the plus sin of the canon s wheel (at the (0,0) position), since we will assume that this represents the startin point of the ball s trajectory. So, for example, select olf ball option, enter 65 for the anle, and 10 for the initial speed. Click on Fire. When the path of the olf ball is displayed, use the tape measurement tool to measure the distance between the startin point (oriin) and the point where the object landed on the x-axis. You should be able to et about 7.8m. Place this value in the table below, and repeat this for the rest of the values iven. A) Projectile Launch Anle versus Rane (for a fixed initial velocity): First, you will investiate how the launch anle impacts the rane of the projectile. For this question, use a fixed initial velocity, such as 10 m/s, and vary the anle. After each trial, measure the rane and record it below. Then, create a scatter plot of the points obtained in the table. Make sure to label the axes of the raph and choose an appropriate scale. Launch anle (derees) =10m/s Rane (m) 15 5 35 45 55 65 7.8 75
Conclusion: Describe how the launch anle and rane are related, makin sure to comment on which anle results in maximum rane. Also comment on if this conclusion arees with your initial uess. B) Initial Velocity versus Rane of the Projectile (for a fixed anle): For this exercise, you will investiate how the initial velocity impacts the rane of the olf ball. Use a fixed anle of 45 derees, and vary the initial velocity as suested below. Fill in the followin table usin the simulator, and then raph the results in the coordinate system provided. Make sure to label the axes and choose an appropriate scale for the axes. Notes: Clear the previous trials by clickin on erase prior to startin this portion. Also, note that you can zoom in or out by clickin on the + or - sins located riht above the fire command. This will help you measure distances more accurately. θ = 45 Initial Velocity(m/s) Rane 5 10 15 0 5 30 Conclusion: (For a fixed anle) As the initial velocity increases, the rane Does initial velocity and rane appear to have a linear or quadratic relationship? How can we tell? 3
C) Launch Anle versus Maximum Heiht Reached (for a fixed initial velocity) For this question, you will investiate how the launch anle affects the maximum heiht reached. Assume a fixed initial velocity, such as 15 m/s. =15 m/s Launch Anle derees) 0 30 40 50 60 70 80 90 Maximum heiht (m) Conclusion: Describe how the launch anle and maximum heiht are related: 3) Checkin the Experimental Results aainst the Theoretical Formulas: A formula from physics relates ives the rane of a projectile as a function of the launch anle and the initial velocity. R sin( ) and θ represents the launch anle in derees. θ = where = 9.8 m/s, is the initial velocity (m/s) A) Launch Anle vs. Rane: In (A) above, you created a table usin empirical data on the relationship between the anle of the projectile and the rane. Let s check one of those values usin the formula. For θ =45 derees, and once aain assumin that =10 m/s, what does the formula ive for the value of R? (Make sure your calculator is in deree mode.) Show work in the space provided. How does this compare this with the value obtained in (A) for a 45 deree launch? R = sin( θ ) 4
B) Initial Velocity vs. Rane: In (B) above, you explored the relationship between initial velocity and rane of the projectile. Now, use the formula iven above to check the case where initial velocity equals 0 m/s. Recall that for this part, we used a fixed anle of 45 derees for each of the trials. How does the formula prediction for R compare with the experimental results? R = sin( θ ) Take a look at the above formula once aain. For a fixed anle, this formula would ive the rane as a function of the initial velocity. Does this formula confirm your conclusions about whether the relationship between the variables is linear or quadratic? Explain. C) Launch Anle versus Maximum Heiht: The formula from physics that relates these two variables is iven on the riht: Use this formula to check one of the entries (θ =40 de.) from your table in (C) above. Recall that we had assumed = 15 m/s for that part. How does the answer obtained from the formula compare with the answer obtained from the experiment (or simulation) in C? H V0 sin = θ Explain, investiatin the above formula, why the hihest value of the maximum heiht is attained with an initial anle of 90 derees. 5
4) Settin the Correct Parameters in Order to Hit a Taret A) In order to hit a taret that is 18 m from the canon, and usin an initial velocity of 0 m/s, what launch anle should be used? Start by measurin a distance that is 18 m from the initial point, and dra the taret there. First use trial-and-error by adjustin the simulator launch anle to et a rouh estimate for the anle that will do the job. Don t waste too much time here, since you will be able to find the exact anle in the next part. Estimated anle: Next, use the followin formula to determine the anle of the projectile that will result in hittin the taret. (= 9.8m/s, represents the initial velocity and R is the rane. Since the unknown is θ, you will need to solve this equation for θ.) R = sin( θ ) B) Determine the initial velocity that one needs to hit a olf ball in order to hit a taret that is 60m away from the initial position. For this question, assume that the anle is 45 derees. You can use the formula introduced above, and check your answer usin the simulator. C) Use the maximum heiht formula to determine the initial velocity for launchin the olf ball, so that the maximum heiht that it reaches equals 5. meters, assumin that the launch anle is 30 derees. Use the iven formula to determine the answer, and then check your answer usin the simulation. To check the answer, you will need to place a taret 5. m. above the x- axis- but where should this taret be placed horizontally? H V0 sin θ = 6
5) Derivation of Some of the Formulas Used: Two main equations for projectile motion are the equations that describe the vertical and the horizontal components of motion. 1 Vertical motion: y = t + V0sin( θ ) t+ y0, where y represents distance from the round t seconds after the projectile was thrown in the air. (For this question y 0 = 0, since initial distance from the round is assumed to be 0.) Horizontal motion: x = V cos( θ ) t 0, where x represents the horizontal distance from the initial point, t seconds after it was launched. A) Determine the time to reach maximum heiht usin the first equation above. Hint: the maximum heiht for a parabola is reached at t = - b/(a), where a is the coef. of the quadratic term, and b is the coef. of the linear term. [Students who have taken calculus can find the maximum by settin dy/dt =0] B) Use your answer from (A) above to derive the formula: H V0 sin θ =, which ives the maximum heiht reached by the projectile. [Basically, all you need to do is to substitute the value of time found above into the vertical motion equation, to find the heiht at that moment in time.] C) Derive the rane formula iven below. To find an equation that ives the rane as a function of initial velocity and the anle theta, you will use the fact that when the object reaches the round, y =0. That will ive us one equation, where you can solve for t. Also, when the object reaches the round, x=r. This will ive you a second equation, where you can also solve for t. Basically, both t s are equal since they both represent total fliht time of the projectile from start to when it hits the round So, set the two t equations equal and solve for R. You will also need to know the double anle formula: sin( θ) = sinθcosθ sin( θ ) R = 7