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Problem Analysis:

A cannonball is fired horizontally with a speed of 25.0 m/s from a height of 53.0 m. The task is to calculate the following:

1. Initial horizontal speed: Given directly as $25.0 \, \text{m/s}$.
2. Initial vertical speed: Since the cannonball is fired horizontally, the vertical speed at the start is $0 \, \text{m/s}$.
3. Time the cannonball remains in the air: Determined by the vertical motion using the formula for free fall.
4. Horizontal distance traveled: Calculated using the horizontal motion formula, knowing the time the ball is in the air.

Step-by-Step Solution:

1. Initial Horizontal Speed:

This is given directly: $v_x = 25.0 \, \text{m/s}.$

2. Initial Vertical Speed:

The vertical speed at launch is $v_y = 0 \, \text{m/s}$.

3. Time in Air ($t$):

The time in air depends on the height of the fall. Using the equation for free fall: $y = \frac{1}{2} g t^2,$ where:

• $y = 53.0 \, \text{m}$,
• $g = 9.8 \, \text{m/s}^2$ (acceleration due to gravity),
• $t$ is the time to solve for.

Rearranging for $t$: $t = \sqrt{\frac{2y}{g}}.$ Substitute the values: $t = \sqrt{\frac{2(53.0)}{9.8}} = \sqrt{\frac{106.0}{9.8}} = \sqrt{10.82} \approx 3.29 \, \text{seconds}.$

So, the ball remains in the air for approximately 3.29 seconds.

4. Horizontal Distance Traveled ($x$):

The horizontal distance is determined using the formula: $x = v_x t,$ where:

• $v_x = 25.0 \, \text{m/s}$,
• $t = 3.29 \, \text{seconds}$.

Substitute the values: $x = 25.0 \times 3.29 = 82.25 \, \text{meters}.$

The ball lands approximately 82.3 meters from the base of the building.

Final Answers:

1. Initial horizontal speed: $25.0 \, \text{m/s}$.
2. Initial vertical speed: $0 \, \text{m/s}$.
3. Time in air: $3.29 \, \text{seconds}$.
4. Horizontal distance: $82.3 \, \text{meters}$.

Would you like further clarification or more detailed explanations?

Related Questions:

1. How does air resistance affect the motion of the projectile in real-life scenarios?
2. What would the horizontal distance be if the initial horizontal speed were doubled?
3. How does changing the height of the cannon affect the time in air?
4. What would happen if the cannonball were launched at an angle instead of horizontally?
5. How does gravity impact the vertical motion compared to the horizontal motion?

Tip:

When solving projectile motion problems, always separate the vertical and horizontal motions and analyze them independently!