A disc is thrown at 21.1 meters per second at a 15.2 degree angle to the horizon. How far away does the disc land? Delta x equals blank meters; fill in the blank.

To determine how far the disc travels horizontally (i.e., \(\Delta x\)), we can break the initial velocity into horizontal and vertical components and use kinematic equations. Here's the step-by-step breakdown:

### Step 1: Break velocity into components
The initial velocity is 21.1 m/s at a \(15.2^\circ\) angle to the horizontal.

- **Horizontal velocity component**: 
  \[
  v_x = v_0 \cdot \cos(\theta) = 21.1 \cdot \cos(15.2^\circ)
  \]

- **Vertical velocity component**: 
  \[
  v_y = v_0 \cdot \sin(\theta) = 21.1 \cdot \sin(15.2^\circ)
  \]

### Step 2: Find the time of flight
The vertical motion is influenced by gravity. The time of flight depends on how long it takes for the disc to rise and fall back to the ground.

Using the kinematic equation for vertical motion:
\[
v_y = v_0 \cdot \sin(\theta) - g \cdot t
\]
At the maximum height, the vertical velocity is zero, and since the object returns to the same height (assuming flat ground), the total time of flight is:
\[
t_{\text{total}} = \frac{2 \cdot v_0 \cdot \sin(\theta)}{g}
\]
where \( g = 9.8 \, \text{m/s}^2 \).

### Step 3: Calculate horizontal distance (\(\Delta x\))
The horizontal distance traveled by the disc is:
\[
\Delta x = v_x \cdot t_{\text{total}}
\]

Now, let's compute the values.The disc lands approximately 22.99 meters away.

Would you like more details or have any questions? 

Here are five related questions to explore:
1. How would the horizontal distance change if the angle was increased to 30 degrees?
2. What is the maximum height the disc reaches during its flight?
3. How long does the disc stay in the air?
4. What effect does air resistance have on the calculated distance?
5. How would a higher initial velocity affect the distance?

**Tip**: The horizontal distance depends primarily on the horizontal velocity and time in the air. Increasing the angle too much can reduce distance due to the lower horizontal speed.

Learn how to calculate the horizontal distance traveled by a disc thrown at 21.1 meters per second at a 15.2-degree angle to the horizon. This step-by-step guide covers concepts in kinematics, projectile motion, and trigonometry. Ideal for students in Grades 9-12.

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