A cannon shoots a ball at an angle θ above the horizontal ground. (a) Neglecting air resistance, use Newton's second law to find the ball's position as a function of time. (Use axes with x measured horizontally and y vertically.) (b) Let r(t) denote the ball's distance from the cannon. What is the largest possible value of θ if r (t) is to increase throughout the ball's flight? [Hint: Using your solution to part (a) you can write down r^2 as x^2 + y^2 , and then find the condition that r^2 is always increasing.]
x(t) = (vicosθ)t
y(t) = (visinθ)t -1/2gt^2
The Attempt at a Solution
While the part for 'a' was a piece of cake (equations in "relevant equations" above), I am having a hard time figuring out the best way to deal with 'b'. From the hint, I was thinking you take x^2 + y^2, take the derivative of it with respect to θ, then set it to 0 to find what values of θ it will be increasing for. However, I tried this and with how many trig values I ended up with in the equation, I am not even sure how to find the zeros for the function, so I can't discern where it will be zero. Does anyone have any suggestions?
In case it was the right approach, taking the derivative of x^2 + y^2 gave me this:
2vi^2t^2sinθcosθ - 2vi^2t^2cosθsinθ - gvit^3cosθ