II) A ball is thrown horizontally from the roof of a building 9.0 m tall and lands 9.5 m from the
base. What was the ball’s initial speed? [GSE 3-32]

To determine the ball's initial speed, we can approach this as a projectile motion problem, where the ball is thrown horizontally from a height.

### Given:
- Height of the building (\( h \)) = 9.0 m
- Horizontal distance (\( d \)) = 9.5 m
- Acceleration due to gravity (\( g \)) = 9.8 m/s² (assuming free-fall conditions)
- The initial vertical velocity (\( v_{0y} \)) = 0 m/s (since it is thrown horizontally)

### Step 1: Find the time taken to fall
The time it takes for the ball to fall from the roof can be calculated using the equation for vertical motion:
\[
h = \frac{1}{2} g t^2
\]
Solving for \( t \):
\[
t = \sqrt{\frac{2h}{g}}
\]
Substitute the known values:
\[
t = \sqrt{\frac{2(9.0)}{9.8}} = \sqrt{\frac{18.0}{9.8}} = \sqrt{1.8367} \approx 1.355 \text{ s}
\]

### Step 2: Calculate the horizontal velocity
Now, we use the horizontal distance traveled to find the initial horizontal speed. The horizontal velocity is constant, so we use the formula:
\[
d = v_0 t
\]
Solving for \( v_0 \):
\[
v_0 = \frac{d}{t}
\]
Substitute the known values:
\[
v_0 = \frac{9.5}{1.355} \approx 7.01 \text{ m/s}
\]

### Final Answer:
The ball’s initial speed is approximately **7.01 m/s**.

Would you like further clarification on this? 

Here are 5 related questions for practice:
1. How would the initial speed change if the building were 15 meters tall?
2. What would happen to the horizontal distance if the initial speed was doubled?
3. How long would it take the ball to hit the ground if thrown from a 12-meter building?
4. What is the final speed of the ball just before it hits the ground?
5. How would air resistance affect the motion of the ball?

**Tip:** In projectile motion problems, horizontal and vertical motions are independent of each other!

A ball is thrown horizontally from the roof of a building 9.0 m tall and lands 9.5 m from the base. What was the ball’s initial speed?

Discover how to calculate the initial speed of a ball thrown horizontally from a 9.0-meter tall building, landing 9.5 meters away. This problem covers key concepts in kinematics, including projectile motion and the independence of horizontal and vertical motion. Suitable for high school students.

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