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: Recall projectile - motion formula

The horizontal range formula for a projectile launched with initial velocity \(v_0\) at an angle \(\theta\) from the ground (assuming no air - resistance) is \(R=\frac{v_0^{2}\sin2\theta}{g}\), where \(g\) is the acceleration due to gravity.

Step2: Analyze the function for maximum range

We want to maximize the function \(y = \sin2\theta\). The range of the sine function is \([- 1,1]\). The maximum value of \(\sin2\theta\) is 1.

Step3: Solve for \(\theta\) when \(\sin2\theta = 1\)

If \(\sin2\theta=1\), then \(2\theta = 90^{\circ}\) (since \(\sin90^{\circ}=1\)). Solving for \(\theta\), we get \(\theta = 45^{\circ}\).

Answer:

The launch angle that causes the foam dart to travel the farthest horizontally (assuming no air - resistance) is \(45^{\circ}\).