A projectile leaves ground level at an angle of \(68^{\circ}\) above the horizontal. As it reaches its maximum height, \(H\), it has traveled a horizontal distance, \(d\), in the same amount of time. What is the ratio \(H / d ?\)

First, let's consider the vertical motion. To analyze the vertical motion, we should find the initial vertical velocity at which the projectile is launched. We can do this using the launch angle and the initial velocity as follows: Let \(v_0\) be the initial velocity (magnitude) and \(\theta\) be the angle above the horizontal, which is given as \(68^{\circ}\). Then, the initial vertical velocity, \(v_{0y}\), can be found using the sine function: \(v_{0y} = v_0 \sin{(\theta)}\) We also know that the maximum height (H) is reached when the vertical velocity becomes zero.

Let \(t_H\) represent the time taken to reach maximum height, and \(g\) be the acceleration due to gravity (assuming it is positive downwards). We can use the following kinematic equation to find \(t_H\): \(v_{0y} - gt_H = 0\) Now, we can substitute the expression for \(v_{0y}\) from Step 1 and solve for \(t_H\): \(t_H = \frac{v_0 \sin{(\theta)}}{g}\)

Now let's consider the horizontal motion. The horizontal velocity, \(v_{0x}\), can be found using the cosine function: \(v_{0x} = v_0 \cos{(\theta)}\) The horizontal distance (d) covered can be found by multiplying the horizontal velocity with the time taken to reach maximum height, which equals to \(t_H\): \(d = v_{0x}\cdot t_H\)

Now we have expressions for both H and d using the equations from Step 2 and Step 3. To find the ratio \(\frac{H}{d}\), we can divide the expressions for H and d: \(\frac{H}{d} = \frac{\frac{v_0^2 \sin^2{(\theta)}}{2g}}{\frac{v_0^2 \sin{(\theta)}\cos{(\theta)}}{g}}\) Now we can simplify the expression: \(\frac{H}{d} = \frac{\sin^2{(\theta)}}{2\sin{(\theta)}\cos{(\theta)}} = \frac{\sin{(\theta)}}{2\cos{(\theta)}}\) Now, we can plug in the given angle \(\theta = 68^{\circ}\): \(\frac{H}{d} = \frac{\sin{(68^{\circ})}}{2\cos{(68^{\circ})}}\) Finally, we can calculate the numerical value of this ratio: \(\frac{H}{d} = \frac{0.8988}{2\cdot 0.3746} \approx 1.20\) Therefore, the ratio \(\frac{H}{d}\) is approximately 1.20.