Up, Up, and Strictly Away

A ball with mass \(m\) is thrown from the origin at speed \(V_{0}\) toward the right on an exotic planet where the strength of gravity is \(g^\prime = \frac{g}{10} = \SI[per-mode=symbol]{1}{\meter\per\second\squared}.\)

Let \(\alpha\) be the largest possible angle such that, for all \(\theta<\alpha\), the distance between the ball and its launch point will be strictly increasing for \(t>0\).

What is \(\tan^{2} \alpha,\) to two decimal places?

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Details and Assumptions:

• \(V_{0}=100 \text{ m/s}.\)
• \(m=1 \text{ kg}.\)

Bonus: Generalize this angle for arbitrary values of \(V_{0}\), \(m\), and \(g^\prime\).