https://r2.erweima.ai/mathbot/image/1724813777979ga1xp5uc.jpg

To solve the problem of finding the components of the initial velocity given in the image, we'll break it down as follows:

### Step 1: Identify the Given Information
From the image, we have:
- The initial velocity magnitude \( v \) is \( 37.0 \, \text{m/s} \).
- The angle \( \theta \) relative to the horizontal axis is \( 32.0^\circ \).

### Step 2: Calculate the Components
The components of the velocity can be calculated using trigonometry:

1. **Horizontal Component (\( v_x \))**:
   \[
   v_x = v \cdot \cos(\theta)
   \]
   
2. **Vertical Component (\( v_y \))**:
   \[
   v_y = v \cdot \sin(\theta)
   \]

### Step 3: Plug in the Values

Let's calculate each component:

\[
v_x = 37.0 \, \text{m/s} \cdot \cos(32.0^\circ)
\]
\[
v_y = 37.0 \, \text{m/s} \cdot \sin(32.0^\circ)
\]

### Step 4: Compute Using a Calculator

I'll now calculate these values.The components of the initial velocity are:

- \( v_x \approx 31.38 \, \text{m/s} \) (horizontal component)
- \( v_y \approx 19.61 \, \text{m/s} \) (vertical component)

Do you want further details or have any questions?

Here are some related questions for further exploration:
1. How would the components change if the angle were increased?
2. What would happen to the components if the velocity magnitude was doubled?
3. How can we calculate the angle given the components of velocity?
4. Why is the horizontal component larger than the vertical component in this scenario?
5. How does the direction of the angle (above or below the horizontal) affect the sign of the components?

**Tip:** Remember that trigonometric functions like sine and cosine are sensitive to the angle's units (degrees vs. radians), so always ensure consistency.

Learn how to find the horizontal (Vx) and vertical (Vy) components of initial velocity using trigonometry. This problem involves calculating these components based on given magnitude and angle. Suitable for students in Grades 10-12.

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