Question

You are riding a broomstick and see a golden ball hovering motionless 30 meters directly in front

Answer 1

You should fly at an angle of $14.0^{\circ}$

Explanation:

The motion of the ball is a free fall motion, so the vertical displacement of the ball (in negative directions) is given by

$y=\frac{1}{2}gt^2$

where

$g=9.8 m/s^2$ is the acceleration due to gravity

t is the time

The horizontal distance you have to cover to catch the ball is

$x = 30 m=d$

The vertical and horizontal displacement form a right triangle, so that we can write

$tan \theta = \frac{y}{x}$

where $\theta$ is the direction of your motion. Substituting,

$tan \theta = \frac{gt^2}{2d}$ (1)

We also know that the horizontal position of the person at time t is

$x(t) = v cos \theta t$

And so, for the time t at which the person catches the ball,

$d=v cos \theta t\\t=\frac{d}{v cos \theta}\\t^2=\frac{d^2}{v^2 cos^2 \theta}$

Substituting this expression into (1),

$tan \theta = \frac{g}{2d} \frac{d^2}{v^2 cos^2 \theta}\\2 tan \theta cos^2 \theta = \frac{gd}{2v^2}$

And using

$tan \theta = \frac{sin \theta}{cos \theta}, 2sin \theta cos \theta = sin(2\theta)$

we find:

$sin (2\theta) = \frac{gd}{v^2}=\frac{(9.8)(30)}{25^2}=0.470\\2\theta = sin^{-1}(0.470)=28.0^{\circ}\\\theta=\frac{28}{2}=14.0^{\circ}$

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