Question

mitch was helping his younger sister use a foam dart launcher that she received for her birthday. they wondered what launch angle would cause the foam dart to travel the farthest horizontally. they thought that launching with a higher angle to the ground would give the dart more time to travel before hitting the ground and would therefore allow the foam dart to travel farther. mitch and his sister set up an experiment in their backyard. they used a protractor to adjust the launcher to five different angles: 15°, 30°, 45°, 60°, and 75°. they launched three foam darts from each angle and used a measuring tape to determine how far each dart traveled horizontally. they then calculated the average for each group and graphed the results. they found that the darts launched at 45° went the farthest, while those launched at 15° and 75° traveled the shortest distance. mitch and his sister shared their results with the friend who had given his sister the launcher.

Explanation:

Step1: Analyze projectile - motion concept

The horizontal range formula for a projectile is $R=\frac{v_{0}^{2}\sin2\theta}{g}$, where $v_{0}$ is the initial velocity, $\theta$ is the launch - angle, and $g$ is the acceleration due to gravity.

Step2: Find the maximum of the range function

The function $y = \sin2\theta$ has a maximum value of 1. This occurs when $2\theta = 90^{\circ}$, so $\theta=45^{\circ}$.

Answer:

The launch angle that causes the foam dart to travel the farthest horizontally is $45^{\circ}$.