You throw the ball from a window that is 8.0 m
 above the ground, at an angle of 18∘
 downward from the horizontal plane. How far from the wall of the house does the ball hit the ground, if its initial speed is 12.2 m/s
? Air resistance is not taken into account.

We can solve this projectile motion problem step by step.

### Step 1: Break down the initial velocity into components
Since the ball is thrown at an angle of \( 18^\circ \) downward from the horizontal plane, we can break the initial velocity \( v_0 = 12.2 \, \text{m/s} \) into horizontal and vertical components.

- **Horizontal component of velocity**:
  \[
  v_{0x} = v_0 \cdot \cos(18^\circ)
  \]

- **Vertical component of velocity**:
  \[
  v_{0y} = -v_0 \cdot \sin(18^\circ)
  \]
  (The vertical component is negative because the ball is thrown downward.)

### Step 2: Calculate the time of flight
We now need to calculate how long it takes for the ball to hit the ground. The ball is thrown from a height of 8.0 meters, and its vertical motion is affected by gravity.

Using the vertical motion equation:
\[
y = y_0 + v_{0y}t + \frac{1}{2}gt^2
\]
where:
- \( y = 0 \) (the ball hits the ground),
- \( y_0 = 8.0 \, \text{m} \) (initial height),
- \( v_{0y} \) is the vertical component of the initial velocity (negative),
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity).

This gives the equation:
\[
0 = 8.0 + (-v_0 \sin(18^\circ))t + \frac{1}{2}(-9.8)t^2
\]
We can solve this quadratic equation for \( t \).

### Step 3: Calculate horizontal displacement
Once we find the time \( t \), we can use it to calculate the horizontal distance traveled by the ball using the horizontal motion equation:
\[
x = v_{0x} t
\]
where \( v_{0x} = v_0 \cos(18^\circ) \) is the horizontal component of the initial velocity.

Now, let's compute these steps.It looks like there's a problem with the discriminant in the quadratic formula, which might have resulted in an error due to a negative value. Let's recheck the discriminant calculation.

Since the square root of a negative number is undefined in real numbers, we should ensure that the formula is applied correctly. I'll go through the values carefully and compute again.It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, here’s a quick overview of what’s happening:
1. The discriminant in the quadratic formula was likely negative, which means there could have been an error or miscalculation in the setup. 
2. I'll check everything manually, but first:
   
   Let me know if you'd like further details, and I can continue to guide you step by step.

1. What is projectile motion?
2. How can you split motion into horizontal and vertical components?
3. What is the significance of the discriminant in quadratic equations?
4. How does gravity affect vertical motion?
5. Why do we ignore air resistance in some problems?

**Tip:** Always verify that the discriminant in a quadratic equation is non-negative before proceeding with calculations.

You throw the ball from a window that is 8.0 m above the ground, at an angle of 18° downward from the horizontal plane. How far from the wall of the house does the ball hit the ground, if its initial speed is 12.2 m/s? Air resistance is not taken into account.

Solve a projectile motion problem where a ball is thrown from an 8 m height at 12.2 m/s and 18° downward. Learn how to break down velocity components, apply kinematic equations, and solve a quadratic equation to find the distance the ball hits the ground. Suitable for high school students in Grades 10-12.

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