Question

Two angles are complementary when their sum is \(90.0^{\circ}\) Find the ranges for two projectiles launched with identical initial speeds of $36.2 \mathrm{m} / \mathrm{s}$ at angles of elevation above the horizontal that are complementary pairs. (a) For one trial, the angles of elevation are \(36.0^{\circ}\) and \(54.0^{\circ} .\) (b) For the second trial, the angles of elevation are \(23.0^{\circ}\) and \(67.0^{\circ} .\) (c) Finally, the angles of elevation are both set to \(45.0^{\circ} .\) (d) What do you notice about the range values for each complementary pair of angles? At which of these angles was the range greatest?

Short answer

Answer: The angle of elevation with the greatest range value in this problem is 45.0°. The significance of complementary pairs in this context is that the sum of the ranges of projectiles launched at complementary angles is constant.

Step-by-step solution

Find the range for Trial (a)

For this trial, the angles of elevation are \(\theta_1 = 36.0^{\circ}\) and \(\theta_2 = 54.0^{\circ}\). Using the range formula, we'll find the respective ranges: \(R_1 = \frac{(36.2)^2 \sin(2\times 36)}{9.81}\) \(R_2 = \frac{(36.2)^2 \sin(2\times 54)}{9.81}\)

Find the range for Trial (b)

For this trial, the angles of elevation are \(\theta_1 = 23.0^{\circ}\) and \(\theta_2 = 67.0^{\circ}\). Using the range formula, \(R_1 = \frac{(36.2)^2 \sin(2\times 23)}{9.81}\) \(R_2 = \frac{(36.2)^2 \sin(2\times 67)}{9.81}\)

Find the range for Trial (c)

For this trial, the angles of elevation are both set to \(\theta_1 = \theta_2 = 45.0^{\circ}\). Thus, \(R_1=R_2\). Using the range formula, \(R_1 = R_2 = \frac{(36.2)^2 \sin(2\times 45)}{9.81}\)

Compare and analyze the range values for each complementary pair of angles

Calculate the range values for each angle in each trial. Observe and compare the range values for each complementary pair. Notice that for each complementary pair, the sum of the ranges is the same.

Determine the angle with the greatest range

Observe and compare the range values for all angles and determine which angle has the greatest range value.