Question

A hunter aims directly at a target (on the same level) 38.0 m away. (a) If the arrow leaves the bow at a speed of 23.1 m/s, by how much will it miss the target? (b) At what angle should the bow be aimed so the target will be hit?

Short answer

The obtained values of the distance and angle are $y=13.27\mathrm{m}$and $\theta =20.05°$, respectively.

Step-by-step solution

Step 1. Calculate the time of flight

In this problem, consider the downward direction as positive y and the point at which the arrow leaves the bow as the origin. In order to determine the angle of launch, use the relation of level horizontal range.

Given data:

The hunter aims at the target at a distance of $d=38\mathrm{m}$.

The speed of the arrow is $v=23.1\mathrm{m}/\mathrm{s}$.

The formula to calculate the time of flight can be written as:

$t=\frac{d}{v}$

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

$\begin{array}{l}t=\left(\frac{38\mathrm{m}}{23.1\mathrm{m}/\mathrm{s}}\right)\\ t=1.645\mathrm{s}\end{array}$

Step 2. Calculate the distance by which the hunter will miss the target

The formula to calculate the distance can be written as:

$y=ut+\frac{1}{2}g{t}^2$.

Here, u is the velocity in the vertical direction whose value is zero, y is the distance by which the hunter misses the target, and g is the gravitational acceleration.

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

$\begin{array}{l}y=\left(0\right)t+\frac{1}{2}\left(9.81 \mathrm{m}/{\mathrm{s}}^2\right){\left(1.645\mathrm{s}\right)}^2\\ y=13.27\mathrm{m}\end{array}$

Thus, $y=13.27\mathrm{m}$is the distance by which the hunter misses the target.

Step 3. Calculate the launch angle

The relation to calculate the launch angle can be written as:

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

Here, $\theta$is the required angle.

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

$\begin{array}{c}38\mathrm{m}=\frac{{\left(23.1\mathrm{m}/\mathrm{s}\right)}^2\sin 2\theta }{\left(9.81\mathrm{m}/{\mathrm{s}}^2\right)}\\ \theta =20.05°\end{array}$

Thus, $\theta =20.05°$is the required launch angle.