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Alice and Bob are having a competition to see who can throw a baseball the farthest fromthe top of a hill which slopes downward uniformly at an angle ? relative to the horizontaldirection. Alice plays baseball, so Bob knows that heâs probably gonna lose. Fortunately,Bob has taken mechanics, and he realizes that there is an optimal angle ? at which he shouldthrow the ball so that it has the greatest range, so maybe he can use physics to give him achance. Attempt to determine that optimal angle as follows:(a) Let the range of the projectile refer to the distance it travels in the horizontal directionbefore hitting the ground. Determine an expression for the range of the projectile asa function of ?. This expression should be a symbolic one that contains ? and ? andperhaps some other relevant parameters like the initial speed v with which the ball isthrown, and the magnitude of the acceleration due to gravity g.(b) Fix the value of ? to be ?/3, choose a reasonable value for v0, and plot the expressionfor the range as a function of ?. You may find it useful to use a programming language(like Mathematica or Python or Matlab) to perform this plotting, and if you do this,you may simply print out the plot. You may also consider finding a website or appwith plotting capabilities to either generate the plot and print it out, or generate theplot and allow you to stetch in on your paper. These days, itâs not so hard to get acomputer to help you plot a function.(c) To check your expression for the range, consider the following limiting cases. In eachcase, forget the math for a moment and attempt to predict what the range expressionshould give in those cases, and then check to see if your answer agrees with yourprediction. If not, you should re-evaluate either your physical intuition that led to theprediction, or your math, or both. In each limiting case, you may assume that all othervariables are held constant when the limit is taken.i. v ? ?ii. v ? 0iii. g ? ?iv. g ? 0v. ? ? 0vi.? ? ?/23vii. ? ? ?/2(d) Use your expression (that you have now hopefully checked to some degree in part (c)),to determine the angle ? which maximizes the range of the ball for a given initial speedand slope angle ?.
