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Cannonball

Let's fire a cannon. Recall the general equation for projectile motion. The height metres of the ball whose horizontal distance is metres away is given by the equation

where , , and . Here, denotes the initial speed, denotes the initial height, denotes the gravitational constant on earth, and denotes the angle of elevation.
The cannon fires from ground level at a initial speed of .

Question 1. Write down the values of and .

Create a GeoGebra simulation given your values of and in the previous part by entering the following commands into the input bar. Press ENTER after each input.
  1. T = Slider(0.1, 89.9, 0.1)
  2. S = T * pi / 180
  3. g = 9.81
  4. a = g / (200 * (cos(S))^2)
  5. b = tan(S)
  6. c = 0
  7. y = a*x^2 + b*x + c

Cannonball

Question 2. Fix . What is the horizontal distance that the cannonball travels?

Question 3. Find a different value of that gives the same horizontal distance.

Question 4. Find the value of that maximises the horizontal distance.

Bonus Question. Why does this happen?