I'm back problem - 3 (Is this just mechanics?)

Calculus Level 5

A ball is thrown from the edge of a cliff $50\: m$ high, at an angle of $30^\circ$ upward from the horizontal and at a speed of $10\text{ m/s}$ . Refer to (2) in the figure. Neglecting air friction and other external factors, determine the length $A$ of the path traveled by the ball (depicted in blue dashed lines). Express your answer as $\lfloor A \rfloor$ .

The answer is 65.

1 solution

Efren Medallo
Oct 22, 2015

The general equation for a parabolic path is

$\large y = x\:tan \:\theta - \frac{g}{2v^{2}\:cos^{2}\:\theta }\:x^{2}$

So, we only have to find the horizontal range of the travel, which is given by

$\large x_{0} = v\:cos \:\theta\cdot (\frac{v}{g}sin\:\theta + \sqrt{\frac{v^{2}\:sin^{2}\:\theta}{g^2}\: + \: \frac{100}{g}})$

Then, this will be the upper limit of the length of the travel path, which is given by

$A = \large \int_0^{x_{0}} \sqrt {1 + (y')^{2} } \: dx$

where $A \approx 65.4945$ , so $\lfloor\: A \: \rfloor = \boxed{65}$

I'm back problem - 3 (Is this just mechanics?)

The answer is 65.

1 solution

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