A ball is thrown horizontally from the top of a 20-m high hill. It strikes the ground at an angle of 45°. With what speed was it thrown? Use g = 10 m/s².

To solve this problem, we can break it down into two components: the horizontal motion and the vertical motion of the ball. 

### Step 1: Analyze the Vertical Motion
The ball is thrown horizontally, so initially, it has no vertical velocity. The only vertical motion comes from the acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \).

Using the equation of motion for vertical displacement:
\[
h = \frac{1}{2} g t^2
\]
where:
- \( h = 20 \, \text{m} \) (height of the hill)
- \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity)
- \( t \) is the time of flight

Rearranging to solve for \( t \):
\[
t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 \times 20}{10}} = \sqrt{4} = 2 \, \text{seconds}
\]

### Step 2: Analyze the Horizontal Motion
Since the ball is thrown horizontally, its horizontal velocity \( u_x \) remains constant throughout the motion. We know that the horizontal velocity \( u_x \) can be related to the total velocity \( u \) and the angle of impact (45° in this case).

The horizontal and vertical components of the velocity at the point of impact are equal because the angle of impact is 45°:
\[
u_x = u_y
\]
where \( u_y \) is the vertical velocity at the moment of impact.

### Step 3: Calculate the Vertical Velocity at Impact
The vertical velocity at impact \( u_y \) can be found using the equation:
\[
u_y = g \cdot t = 10 \, \text{m/s}^2 \times 2 \, \text{seconds} = 20 \, \text{m/s}
\]

### Step 4: Determine the Horizontal Velocity (Initial Speed)
Since \( u_x = u_y \) at the point of impact, we have:
\[
u_x = 20 \, \text{m/s}
\]

This \( u_x \) is the speed with which the ball was thrown, so the ball was thrown with a speed of \( 20 \, \text{m/s} \).

Would you like more details or have any questions about the solution?

Here are five related questions to further your understanding:
1. What would be the speed of the ball if it were thrown from a different height?
2. How would the impact angle change if the ball was thrown with an initial upward velocity?
3. What is the range of the ball's horizontal displacement from the point of release to where it strikes the ground?
4. How would air resistance affect the time of flight and the impact velocity?
5. If the ball was thrown from a moving platform, how would the platform's velocity affect the ball's trajectory?

**Tip:** When analyzing projectile motion, always break the motion into horizontal and vertical components to simplify your calculations.

Discover how to calculate the initial speed of a ball thrown horizontally from a 20-m high hill, striking the ground at a 45° angle. This problem covers key concepts in projectile motion, kinematics, and trigonometry. Perfect for high school students in Grades 9-12.

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