https://file.aiquickdraw.com/mathbot/file/17274090742940y7m55s2.jpg

Let's go through the problem based on the provided image:

### Problem Analysis
1. **Initial Information:**
   - A ball is kicked at an angle of \( 30^\circ \) with an initial velocity of \( v_0 = 10 \, \text{m/s} \).
   - The trajectory forms a parabolic path, as depicted in the graph.

2. **Given Questions:**
   - **(4)** Find the initial velocity components along the x-axis and y-axis.
   - **(5)** Calculate the speed of the ball at point A.
   - **(6)** Determine the speed of the ball at \( t = 0.5 \, \text{s} \).
   - **(7)** Calculate the maximum height reached by the ball.
   - **(8)** Find the farthest distance reached by the ball (horizontal range).

### Solution Approach

1. **Initial Velocity Components:**
   - Horizontal component: \( v_{0x} = v_0 \cos \theta \).
   - Vertical component: \( v_{0y} = v_0 \sin \theta \).

2. **Velocity at Point A:**
   - Point A is at the peak, so the vertical component \( v_y = 0 \).
   - Speed at A is only due to the horizontal component.

3. **Velocity at \( t = 0.5 \, \text{s} \):**
   - Use the equations of motion to find \( v_x \) and \( v_y \) at \( t = 0.5 \, \text{s} \).

4. **Maximum Height:**
   - Use the formula \( h_{\text{max}} = \frac{(v_{0y})^2}{2g} \).

5. **Horizontal Range:**
   - Use the formula \( R = \frac{v_0^2 \sin 2\theta}{g} \).

Let's calculate these values step-by-step.### Calculated Values

1. **Initial Velocity Components:**
   - Horizontal component (\( v_{0x} \)): \( 8.66 \, \text{m/s} \).
   - Vertical component (\( v_{0y} \)): \( 5.00 \, \text{m/s} \).

2. **Speed at Point A:**
   - The speed at point A is \( 8.66 \, \text{m/s} \) (since the vertical component is zero at the peak).

3. **Speed at \( t = 0.5 \, \text{s} \):**
   - The speed of the ball at \( t = 0.5 \, \text{s} \) is approximately \( 8.66 \, \text{m/s} \).

4. **Maximum Height:**
   - The maximum height reached by the ball is \( 1.28 \, \text{m} \).

5. **Horizontal Range:**
   - The farthest distance reached by the ball (horizontal range) is \( 8.84 \, \text{m} \).

Would you like more details on any of these calculations, or do you have any other questions?

### Follow-up Questions:
1. How would the results change if the initial velocity was increased?
2. What would be the time to reach the maximum height?
3. How long does the ball stay in the air before hitting the ground?
4. How would the range change if the angle was \(45^\circ\)?
5. What would be the impact of a different gravitational acceleration, like on the Moon?

**Tip:** For projectile motion problems, always break down the initial velocity into horizontal and vertical components first!

Calculate various properties of a ball's motion when kicked at an angle with an initial speed.

Explore the detailed calculations involved in projectile motion when a ball is kicked at a 30-degree angle with an initial speed of 10 m/s. Understand concepts such as initial velocity components, maximum height, time of flight, and total distance covered.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation