Question

William Tell must split the apple on top of his son’s head from a distance of 27 m. When William aims directly at the apple, the arrow is horizontal. At what angle should he aim the arrow to hit the apple if the arrow travels at a speed of 35 m/s?

Short answer

The obtained value of the angle is $\theta =6.2°$.

Step-by-step solution

Step 1. Level horizontal range formula

In this problem, the heights of the arrow at the beginning and end are equal; so the formula of horizontal range can be used to determine the angle. Here, the range and the speed of the arrow are already given.

Given data:

The distance from which the apple is split is $d=27\mathrm{m}$.

The speed of the arrow is $v=35\mathrm{m}/\mathrm{s}$.

The formula of level horizontal range can be written as:

$R=\frac{v^2\sin 2\theta }{g}$

Here, $\theta$is the required angle, R is the range whose value is $d=27\mathrm{m}$, and g is the gravitational acceleration.

Step 2. Determine the angle at which William should aim the arrow

On plugging the values in the above relation, you get:

$\begin{array}{c}27\mathrm{m}=\frac{{\left(35\mathrm{m}/\mathrm{s}\right)}^2\sin 2\theta }{\left(9.81\mathrm{m}/{\mathrm{s}}^2\right)}\\ \sin 2\theta =0.216\\ \theta =6.2°\end{array}$

Thus, the required angle is $\theta =6.2°$.