This problem involves projectile motion. The ball is launched with an initial speed of 291 m/s at a 45° angle, and we need to determine three quantities: the maximum height $H$, the total time of flight $t$, and the horizontal range $R$.

Given:

• Launch speed: $v_0 = 291 \, \text{m/s}$
• Launch angle: $\theta = 45.0^\circ$
• Acceleration due to gravity: $g = 9.81 \, \text{m/s}^2$

1. Maximum Height (H):

The formula for the maximum height $H$ reached by a projectile is given by:

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

Since the launch angle is 45°, we can calculate the vertical component of the initial velocity:

$v_{0y} = v_0 \sin(\theta) = 291 \, \text{m/s} \times \sin(45^\circ) = 291 \, \text{m/s} \times \frac{\sqrt{2}}{2} \approx 205.5 \, \text{m/s}$

Now substitute into the height formula:

$H = \frac{(205.5)^2}{2 \times 9.81} = \frac{42240.25}{19.62} \approx 2155.2 \, \text{m}$

2. Time of Flight (t):

The total time of flight for a projectile is given by:

$t = \frac{2 v_0 \sin(\theta)}{g}$

Substitute the values:

$t = \frac{2 \times 291 \, \text{m/s} \times \sin(45^\circ)}{9.81} = \frac{2 \times 291 \, \text{m/s} \times \frac{\sqrt{2}}{2}}{9.81} \approx \frac{2 \times 205.5}{9.81} \approx \frac{411}{9.81} \approx 41.9 \, \text{s}$

3. Horizontal Range (R):

The horizontal range of the projectile is given by:

$R = v_0 \cos(\theta) \times t$

First, calculate the horizontal component of the velocity:

$v_{0x} = v_0 \cos(\theta) = 291 \, \text{m/s} \times \cos(45^\circ) = 291 \, \text{m/s} \times \frac{\sqrt{2}}{2} \approx 205.5 \, \text{m/s}$

Now calculate the range:

$R = 205.5 \, \text{m/s} \times 41.9 \, \text{s} \approx 8613.45 \, \text{m}$

Final Answers:

• The maximum height $H$ reached by the ball is approximately 2155.2 m.
• The total time of flight $t$ is approximately 41.9 seconds.
• The horizontal range $R$ of the cannon is approximately 8613.45 meters.

Would you like more details or have any questions about these calculations?

Additional questions:

1. How does the launch angle affect the height and range of the projectile?
2. What happens if the angle is increased or decreased?
3. How would the results change if the ball was launched with a different initial speed?
4. How does air resistance affect the actual trajectory and results?
5. Can you calculate the velocity components at the highest point of the trajectory?

Tip: Always remember that the horizontal and vertical components of projectile motion are independent of each other. The time of flight depends solely on the vertical motion, while the range depends on the horizontal motion.