We are given the initial speed $v_0 = 190 \, \text{m/s}$, the angle of elevation $\theta = 60^\circ$, and the initial height of 80 meters. We also know the acceleration due to gravity $g = 9.8 \, \text{m/s}^2$. Let's solve this step by step for each part of the question.

(a) Find the range of the projectile

The range $R$ is the horizontal distance the projectile travels before hitting the ground. The formula for the range of a projectile launched from a height is:

$R = \frac{v_0 \cos(\theta)}{g} \left( v_0 \sin(\theta) + \sqrt{(v_0 \sin(\theta))^2 + 2gh_0} \right)$

Where:

• $v_0 = 190 \, \text{m/s}$ is the initial speed,
• $\theta = 60^\circ$ is the angle of launch,
• $g = 9.8 \, \text{m/s}^2$ is the acceleration due to gravity,
• $h_0 = 80 \, \text{m}$ is the initial height.

(b) Find the maximum height

The maximum height $H$ can be found using the formula:

$H = h_0 + \frac{(v_0 \sin(\theta))^2}{2g}$

(c) Find the speed at impact

The speed at impact can be found by using energy conservation principles or by calculating the final velocities in both the horizontal and vertical directions. The speed $v_f$ at impact is:

$v_f = \sqrt{v_x^2 + v_y^2}$

Where:

• $v_x = v_0 \cos(\theta)$ is the horizontal velocity (which remains constant),
• $v_y$ is the vertical velocity at impact, which can be found using kinematic equations.

Now let's calculate each part.It seems like I can’t do more advanced data analysis right now. Please try again later. Let me know if you'd like to go over the steps to manually calculate the results, or if you'd like further assistance on anything else!

Here are five related questions that could expand on the topic:

1. How does changing the launch angle affect the range of a projectile?
2. How does the initial height affect the time of flight for a projectile?
3. What happens to the maximum height if the projectile's initial speed increases?
4. How would air resistance affect the projectile's motion?
5. How can we determine the angle that gives the maximum range for a given initial speed?

Tip: For projectile problems, always resolve the initial velocity into horizontal and vertical components for easier calculation of range and height.