8. Sweden and Norway are at war over who makes the best pickled herring. You are...

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8. Sweden and Norway are at war over who makes the best pickled herring. You are in charge of a Swedish artillery unit that is trying to shoot at some Norwegians located 1000 meters away at the same height as you are. If we neglect the friction of the air, the cannon ball follows a parabolic trajectory; as you certainly recall from prior courses, the cannon ball travels the distance L= V² sin (2a) 9 where V is the initial velocity, a is the angle of the cannon barrel, and g = 9.8 m/s is the gravitational acceleration. You realize that you only have two options for how to shoot. Option (I) is to use 1 bag of gun powder, which produces an initial velocity of exactly V₁ = 100 m/s, and to try to set the angle to QI,nominal = 40°. Option (II) is to use 2 bags of gun powder, which produces exactly VII = 140 m/s, and to try to set the angle to aII,nominal = 15⁰. (a) If you set the angles exactly to the nominal ones, which option is the best? Hint: This is trivial - just calculate the lengths L₁ and L₁ of each shot. The problem is that the cannon is old with a rusty mechanism - you can't set the angle a to exactly the value you want. You can model the actual angle a as a random variable that is normally distributed around anominal with a standard deviation of 1º. Under this scenario, which option is the best? We can solve this problem in 2 ways: (b) First solve it with pen-and-paper, by using the approximate formulas for the mean and standard deviation of a nonlinear function of a random variable (the “error-propagation formula”). (c) Next solve it using numerical simulation. For each option (I and II), create N = 10000 random angles and compute the value of L for each random angle. Suitable Matlab code would look something like V = 100; alpha pi/180* ( 40 + 1*randn (10000, 1) ) ; L = V^2 * sin(2*alpha) / 9.8 ; Now that you have N samples of L, you can plot them in a histogram, and also compute the sample mean and sample standard deviation. (d) Compare the answers from parts (b) and (c). Which approach ("error-propagation formula" or random sampling) do you think is the more accurate? What are the errors in each approach (explain in words)? (e) Compare the conclusions (i.e., which of options I or II is best) in parts (a) and (b-c). Is the answer clear in parts (b-c)? 8. Sweden and Norway are at war over who makes the best pickled herring. You are in charge of a Swedish artillery unit that is trying to shoot at some Norwegians located 1000 meters away at the same height as you are. If we neglect the friction of the air, the cannon ball follows a parabolic trajectory; as you certainly recall from prior courses, the cannon ball travels the distance L= V² sin (2a) 9 where V is the initial velocity, a is the angle of the cannon barrel, and g = 9.8 m/s is the gravitational acceleration. You realize that you only have two options for how to shoot. Option (I) is to use 1 bag of gun powder, which produces an initial velocity of exactly V₁ = 100 m/s, and to try to set the angle to QI,nominal = 40°. Option (II) is to use 2 bags of gun powder, which produces exactly VII = 140 m/s, and to try to set the angle to aII,nominal = 15⁰. (a) If you set the angles exactly to the nominal ones, which option is the best? Hint: This is trivial - just calculate the lengths L₁ and L₁ of each shot. The problem is that the cannon is old with a rusty mechanism - you can't set the angle a to exactly the value you want. You can model the actual angle a as a random variable that is normally distributed around anominal with a standard deviation of 1º. Under this scenario, which option is the best? We can solve this problem in 2 ways: (b) First solve it with pen-and-paper, by using the approximate formulas for the mean and standard deviation of a nonlinear function of a random variable (the “error-propagation formula”). (c) Next solve it using numerical simulation. For each option (I and II), create N = 10000 random angles and compute the value of L for each random angle. Suitable Matlab code would look something like V = 100; alpha pi/180* ( 40 + 1*randn (10000, 1) ) ; L = V^2 * sin(2*alpha) / 9.8 ; Now that you have N samples of L, you can plot them in a histogram, and also compute the sample mean and sample standard deviation. (d) Compare the answers from parts (b) and (c). Which approach ("error-propagation formula" or random sampling) do you think is the more accurate? What are the errors in each approach (explain in words)? (e) Compare the conclusions (i.e., which of options I or II is best) in parts (a) and (b-c). Is the answer clear in parts (b-c)?

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Solutions Step 1 The question presents a scenario where you need to fire a cannonball in a parabolic trajectory to hit a target using two different op... View the full answer