Question

1. The bungee jump company suggests that the standard jump will consist of 10 ‘bounces’ which should take approximately 60 seconds. (Although the jumper will still be in motion at this point, the jump will be considered to be over, and the the jumper will be gently raised all the way back onto the platform above.) Do your model results agree with this timing: 10 bounces in around 60 seconds? 2. The ‘thrill factor’ of bungee jumping is partly determined by the maximum speed experienced by the jumper. What is this maximum speed and when does it occur in relation to the overall jump? Answer this question graphically by plotting the jumper’s velocity against time. 3. Another factor for thrill-seekers is the maximum acceleration experienced by the jumper. More acceleration equals bigger thrills, but too much acceleration can be dangerous. The bungee jump company boasts that the jumper will experience acceleration ‘up to 2g’. Use numerical differentiation to find the acceleration predicted by your model, and plot the jumper’s acceleration against time. What is this maximum acceleration and when does it occur in relation to the overall jump? Is the claim of ‘up to 2g’ acceleration supported by the model? 4. For the writing of promotional material it is of interest to know how far the jumper actually travels in the 60 second jump. One way to answer this question is to compute the integral ?[0 to 60] |v| dt . Use numerical integration to compute this integral and hence determine how far the jumper travels. 5. Part of the proposal is to have a camera installed on the bridge deck, at height D from the water. As the jumper first passes this point, the camera would take a photo which could then be offered for purchase afterwards. It is hoped that the model can provide sufficiently accurate results that the camera could be set to trigger at a predetermined time, for a given set of model parameters. The distance the jumper falls from the platform to the deck is H - D. Hence you need to compute an accurate value for t such that y(t) = H - D. Since you only know y(t) as a discrete set of points, not as a function, you will need to fit an interpolating polynomial. Construct an interpolating polynomial p(t) through the nearest four values of your numerical solution that lie either side of H - D. That is, write MATLAB code to find values yi, yi+1, yi+2, yi+3 such that yi, yi+1 < H - D and yi+2, yi+3 > H - D. Use a rootfinding method of your choice to find the value of t such that p(t) = H - D. Hence, for the model parameters provided, at what time should the camera trigger in order to capture the image of the jumper? 6. The bungee jump company has suggested a ‘water touch’ option could be considered, whereby the jumper just touches the water at the bottom of the first bounce. For the given parameters and assuming a jumper of height 1.75m, how close does the jumper come to touching the water? Investigate how the bungee rope could be altered (its length, its spring constant, or both) to produce a true water touch experience for an 80kg jumper, while keeping as close as possible to 10 bounces in 60 seconds. Note: any combination of parameters that produces acceleration of greater than 2g must be rejected as too dangerous.

1. The bungee jump company suggests that the standard jump will consist of 10 ‘bounces’ which should take approximately 60 seconds. (Although the jumper will still be in motion at this point, the jump will be considered to be over, and the the jumper will be gently raised all the way back onto the platform above.) Do your model results agree with this timing: 10 bounces in around 60 seconds?

2. The ‘thrill factor’ of bungee jumping is partly determined by the maximum speed experienced by the jumper. What is this maximum speed and when does it occur in relation to the overall jump? Answer this question graphically by plotting the jumper’s velocity against time.
3. Another factor for thrill-seekers is the maximum acceleration experienced by the jumper. More acceleration equals bigger thrills, but too much acceleration can be dangerous. The bungee jump company boasts that the jumper will experience acceleration ‘up to 2g’. Use numerical differentiation to find the acceleration predicted by your model, and plot the jumper’s acceleration against time. What is this maximum acceleration and when does it occur in relation to the overall jump? Is the claim of ‘up to 2g’ acceleration supported by the model?
4. For the writing of promotional material it is of interest to know how far the jumper actually travels in the 60 second jump. One way to answer this question is to compute the integral
?[0 to 60] |v| dt .
Use numerical integration to compute this integral and hence determine how far the jumper travels.
5. Part of the proposal is to have a camera installed on the bridge deck, at height D from the water. As the jumper first passes this point, the camera would take a photo which could then be offered for purchase afterwards. It is hoped that the model can provide sufficiently accurate results that the camera could be set to trigger at a predetermined time, for a given set of model parameters.
The distance the jumper falls from the platform to the deck is H - D. Hence you need to compute an accurate value for t such that y(t) = H - D. Since you only know y(t) as a discrete set of points, not as a function, you will need to fit an interpolating polynomial. Construct an interpolating polynomial p(t) through the nearest four values of your numerical solution that lie either side of H - D. That is, write MATLAB code to find values yi, yi+1, yi+2, yi+3 such that yi, yi+1 < H - D and yi+2, yi+3 > H - D. Use a rootfinding method of your choice to find the value of t such that p(t) = H - D. Hence, for the model parameters provided, at what time should the camera trigger in order to capture the image of the jumper?
6. The bungee jump company has suggested a ‘water touch’ option could be considered, whereby the jumper just touches the water at the bottom of the first bounce. For the given parameters and assuming a jumper of height 1.75m, how close does the jumper come to touching the water? Investigate how the bungee rope could be altered (its length, its spring constant, or both) to produce a true water touch experience for an 80kg jumper, while keeping as close as possible to 10 bounces in 60 seconds. Note: any combination of parameters that produces acceleration of greater than 2g must be rejected as too dangerous.

1. The bungee jump company suggests that the standard jump will consist of 10 ‘bounces’ which should take approximately 60 seconds. (Although the jumper will still be in motion at this point, the jump will be considered to be over, and the the jumper will be gently raised all the way back onto the platform above.) Do your model results agree with this timing: 10 bounces in around 60 seconds?

2. The ‘thrill factor’ of bungee jumping is partly determined by the maximum speed experienced by the jumper. What is this maximum speed and when does it occur in relation to the overall jump? Answer this question graphically by plotting the jumper’s velocity against time.
3. Another factor for thrill-seekers is the maximum acceleration experienced by the jumper. More acceleration equals bigger thrills, but too much acceleration can be dangerous. The bungee jump company boasts that the jumper will experience acceleration ‘up to 2g’. Use numerical differentiation to find the acceleration predicted by your model, and plot the jumper’s acceleration against time. What is this maximum acceleration and when does it occur in relation to the overall jump? Is the claim of ‘up to 2g’ acceleration supported by the model?
4. For the writing of promotional material it is of interest to know how far the jumper actually travels in the 60 second jump. One way to answer this question is to compute the integral
?[0 to 60] |v| dt .
Use numerical integration to compute this integral and hence determine how far the jumper travels.
5. Part of the proposal is to have a camera installed on the bridge deck, at height D from the water. As the jumper first passes this point, the camera would take a photo which could then be offered for purchase afterwards. It is hoped that the model can provide sufficiently accurate results that the camera could be set to trigger at a predetermined time, for a given set of model parameters.
The distance the jumper falls from the platform to the deck is H - D. Hence you need to compute an accurate value for t such that y(t) = H - D. Since you only know y(t) as a discrete set of points, not as a function, you will need to fit an interpolating polynomial. Construct an interpolating polynomial p(t) through the nearest four values of your numerical solution that lie either side of H - D. That is, write MATLAB code to find values yi, yi+1, yi+2, yi+3 such that yi, yi+1 < H - D and yi+2, yi+3 > H - D. Use a rootfinding method of your choice to find the value of t such that p(t) = H - D. Hence, for the model parameters provided, at what time should the camera trigger in order to capture the image of the jumper?
6. The bungee jump company has suggested a ‘water touch’ option could be considered, whereby the jumper just touches the water at the bottom of the first bounce. For the given parameters and assuming a jumper of height 1.75m, how close does the jumper come to touching the water? Investigate how the bungee rope could be altered (its length, its spring constant, or both) to produce a true water touch experience for an 80kg jumper, while keeping as close as possible to 10 bounces in 60 seconds. Note: any combination of parameters that produces acceleration of greater than 2g must be rejected as too dangerous.

Added by Mary B.

Below I have marked questions that need answered with _______ and there are 6 questions at the bottom that need answers. Record the total flight time of the cannonball. Answer: 1.01 s Also, record the range of the cannonball. Answer: 10.1 m Next, while leaving the height and launch angle constant, you will be varying the launch speed and noticing the effect on the cannonball’s flight. Before running any trials, predict the relationships between the launch speed and the flight time and between the launch speed and the range of the object fired horizontally. As the launch speed is increased, will the flight time increase, decrease, or remain the same? Answer: same As the launch speed is increased, will the range increase, decrease, or remain the same? Answer: increase Now, run the simulation at the launch speeds indicated and complete the table below. Remember to keep the launch height at 5 m and the launch angle at 0°. (If the projectile lands too close to the right side of the screen to be able to use the scope, zoom out using the magnifying glass buttons at the top left of the screen.) Table 1. Effect of Launch Speed on a Projectile Launched Horizontally. Launch Speed (m/s) Total Flight Time (s) Range (m) 10 1.01 10.1 15 1 15 20 1.01 20.19 25 1.01 25.24 30 1.01 30.29 Do your results match your predictions? Answer: yes Step 7. To help understand your results, consider the case when the launch speed is zero. This would correspond to one-dimensional freefall for an object dropped from rest. Use a kinematics equation to calculate the time it takes an object dropped from rest to fall a distance of 5 m. Answer: _______ Now, write a statement to explain the pattern you found in the flight times recorded in this experiment. Answer: ________ Part B: Object Launched at an Angle The purpose of this portion of the laboratory exercise is to use the PhET simulation to visualize a typical projectile motion problem. Step 1. Reset the PhET simulation by clicking on the yellow eraser button to the left of the red launch button. Now, set the cannon to a height of 0 m and a launch angle of 35°. Don’t worry about the launch speed just yet. Click the button showing a magnifying glass with a plus sign to zoom in. Step 2. Use the tape measure tool to measure the horizontal distance between the cannonball’s launch point and Michelangelo’s David statue. To use the tape measure tool, click and drag it to the base of the cannon. Match the crosshairs on the tape measure to the crosshairs on the cannon. Then, click and drag the right side of the tool to expand the tape measure to reach from the cannon to the bottom of David. Record the horizontal distance from the launch point to the statue. Answer: _______ Step 3. Use the tape measure to measure the vertical height of the statue of David. Answer: _______ Step 4. Use the kinematics equations, and what you know about projectile motion to calculate the answer to this projectile motion problem: At what speed must the cannonball be launched so that it just passes over David’s head? What is the flight time of the cannonball to this point? (Hints: The cannonball may not pass over David’s head at the exact peak of its trajectory. You will have to solve a system of equations to find the solution.) Answer: _______ Step 5. Now, in the PhET simulation, adjust the launch speed to approximately the speed you calculated. Does the cannonball pass directly over David’s head? Answer: _______ If not, go back to your calculations and find your mistake. Part C: Maximum Range of an Object in Projectile Motion The purpose of this portion of the laboratory exercise is to discern both experimentally and algebraically the launch angle that corresponds to the maximum range of a projectile. Step 1. Reset the simulation and zoom back out to the standard view. Set the cannonball’s height to zero, launch speed to 15 m/s, and launch angle to 25°. Fire the cannon and use either the tape measure or the scope tool to determine the range of the projectile. Then, while leaving the height and launch speed constant, adjust the angle to each of the values indicated and complete the table below. Table 2. Dependence of Projectile Range on the Angle of Projection Angle of Projection (°) Range (m) 25 17.57 30 19.49 40 22.59 45 22.99 50 22.59 60 19.86 70 14.74 Step 2. Using Excel or Google Sheets, create a graph of range (y-axis) versus angle of projection (x-axis) based on the data in the table above. Include a title for your graph as well as axis labels with units. Fit a trendline to your data. What type of trendline best fits your data, and what is the equation of this trendline? Answer: _______ Step 3. Using your trendline, find the launch angle that corresponds to the maximum range. What is this launch angle? Answer: _______ Step 4. Let us next show that this launch angle for maximum range will be the same for all projectiles, regardless of launch speed (as long as air resistance is ignored). First, we know that the launch velocity, v0, can be resolved into its x and y components. v0x=v0cos𝜃 and v0y=v0sin𝜃 Consider a projectile that is launched from ground level and returns back to the ground. What is the vertical displacement, 𝑦 − 𝑦0, of the projectile in this case? Answer: _______ Use the kinematics equation for the vertical component of an object in projectile motion y-y0=v0yt-1/2 gt2 to find an equation for the flight time of the projectile from the instant it is launched to the instant it hits the ground. Leave your equation in variable form. Do not plug in any numbers. Answer: _______ Next, combine this equation with the kinematics equation for the horizontal component of an object in projectile motion x-x0=v0xt. Replace x-x0 with 𝑅 (for range) and be sure to plug in the equation above for v0x. Your final answer should correlate the range to the launch velocity, v0, launch angle, 𝜃, and gravitational acceleration constant, 𝑔. Answer: _______ Use the trig identity 2𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃 = 𝑠𝑖𝑛(2𝜃) to simplify your equation. Answer: _______ Now, you should see that if the launch velocity and the gravitational acceleration are held constant, your range will be maximized when 𝑠𝑖𝑛(2𝜃) is maximized. For what value of 𝜃 is 𝑠𝑖𝑛(2𝜃) maximized? (Recall that a sine function varies between -1 and 1; What must 𝜃 be so that 𝑠𝑖𝑛(2𝜃) = 1?) Answer: _______ Does this value for the angle corresponding to maximum range agree with your experimentally determined angle above? Answer: _______ Part D: Symmetry of Launch Angles and Ranges The purpose of this portion of the laboratory exercise is to determine why two launch angles may give the same range. Step 1. Re-examine Table 2 above and your corresponding graph. Were there any pairs of launch angles that resulted in the same range? If so, what are the pairs? Answer:_______ Based on your observations, write a rule that describes when two launch angles will result in the same range. Answer: _______ Step 2. Next, you will want to show that this rule holds for any launch speed. Begin with the last equation you found in Part C. Recopy that equation here. Answer: _______ Revisit the rule you wrote in step 1. You should have found that when the launch angle is symmetric about 45°, the ranges are the same. Another way to state this is that when 𝜃 = 45° + 𝛥𝜃, the range is the same as when 𝜃 = 45° − 𝛥𝜃, where 𝛥𝜃 is any angle between 0 and 45°. Plug 𝜃 = 45° + 𝛥𝜃 into the equation above. Simplify the equation as much as you can, while still leaving it in variable form (i.e. don’t plug any numbers in for v0 or 𝑔). (You may want to use the trig identity 𝑠𝑖𝑛(90° + 𝜙) = 𝑐𝑜𝑠𝜙 which stems from the fact that sine and cosine functions are the same, but shifted 90°.) Answer: _______ Also, plug 𝜃 = 45° − 𝛥𝜃 into your range equation. Again, simplify as much as you can while still leaving the equation in variable form. (This time, you may want to use the trig identity above, plus the additional identity that 𝑐𝑜𝑠(−𝜙) = 𝑐𝑜𝑠𝜙 which reflects the fact that the cosine function is symmetric about the y axis.) Answer: _______ Compare these two equations. What conclusion can you draw? Answer: _______ Questions 1. How does the vertical portion of projectile motion compare to one-dimensional free fall? 2. How does adding air resistance affect the motion of a projectile? What differences in the projectile’s path, height, and range do you observe? Experiment with the PhET simulation to draw specific conclusions. Systematically compare cannonball flights with the same initial conditions but with and without air resistance. 3. With all other initial conditions held constant, how does increasing the launch height affect range? 4. With all other initial conditions held constant, how does increasing launch speed affect maximum height? 5. With all other initial conditions held constant, how does increasing the launch angle affect range? 6. With all other initial conditions held constant, how does increasing the launch angle affect flight time?

Umar Sohail Q.

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