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, a cannon ball travels a distance : L = V2 sin(2 alpha)/g
where V is the initial velocity, 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 alpha1,nominal = 40 0. Option (II) is to use 2 bags of gun powder, which produces exactly V2 = 140 m/s, and to try to set the angle to alpha2,nominal 15 0.
Now we use a new measuring instrument that is not perfect. It causes an uncertainty (i. e., error) in each measurement. Therefore, the traveling distance can be described according to an equation below, where error is a uniformly distributed random variable between [0,5].
A. If you set the angles exactly to the nominal ones, which option is the best? Hint: This is trivial - just calculate the lengths Li and LII 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 αnominal with a standard deviation of 1°. Under this scenario, which option is the best?
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·= V2 * 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)?