.A soccer ball is kicked at 17.5 m/s at an angle of 30° above the horizontal. a. How
long (time) is the ball in the air? b) What is the maximum height it reaches? c)
How far away does the ball land if no one intercepts it?

To solve this problem, we need to break down the initial velocity of the soccer ball into its horizontal and vertical components and use kinematic equations to determine the time of flight, maximum height, and range.

### Given Data:
- Initial velocity (\(v_0\)) = 17.5 m/s
- Angle (\(\theta\)) = 30°
- Acceleration due to gravity (\(g\)) = 9.8 m/s²

### Components of Initial Velocity:
The horizontal (\(v_{0x}\)) and vertical (\(v_{0y}\)) components of the initial velocity can be calculated using trigonometry:

\[ v_{0x} = v_0 \cos(\theta) \]
\[ v_{0y} = v_0 \sin(\theta) \]

Substituting the values:
\[ v_{0x} = 17.5 \cos(30^\circ) = 17.5 \times \frac{\sqrt{3}}{2} \approx 15.14 \, \text{m/s} \]
\[ v_{0y} = 17.5 \sin(30^\circ) = 17.5 \times \frac{1}{2} = 8.75 \, \text{m/s} \]

### (a) Time of Flight:
The time of flight (\(T\)) can be found by determining the time it takes for the ball to reach the peak of its trajectory and then double it (as the descent time is equal to the ascent time).

The time to reach the maximum height (\(t_{peak}\)) is given by:
\[ t_{peak} = \frac{v_{0y}}{g} \]
\[ t_{peak} = \frac{8.75}{9.8} \approx 0.89 \, \text{s} \]

Thus, the total time of flight (\(T\)) is:
\[ T = 2 \times t_{peak} \approx 2 \times 0.89 \approx 1.78 \, \text{s} \]

### (b) Maximum Height:
The maximum height (\(H\)) can be found using the following kinematic equation:
\[ H = v_{0y} t_{peak} - \frac{1}{2} g t_{peak}^2 \]
\[ H = 8.75 \times 0.89 - \frac{1}{2} \times 9.8 \times (0.89)^2 \]
\[ H \approx 7.79 - 3.88 \approx 3.91 \, \text{m} \]

### (c) Range:
The horizontal range (\(R\)) can be calculated using the total time of flight and the horizontal component of the initial velocity:
\[ R = v_{0x} \times T \]
\[ R = 15.14 \times 1.78 \approx 26.94 \, \text{m} \]

### Summary:
a) The ball is in the air for approximately **1.78 seconds**.

b) The maximum height it reaches is approximately **3.91 meters**.

c) The ball lands approximately **26.94 meters** away from the kicking point.

Would you like more details or have any questions? Here are some more problems you might be interested in:

1. How do you determine the horizontal range of a projectile launched from a height?
2. What is the effect of air resistance on projectile motion?
3. How to find the initial velocity if only the range and angle are given?
4. How do you calculate the time of flight for a projectile launched at an angle from an elevated position?
5. How does varying the launch angle affect the range of a projectile?

**Tip:** When solving projectile motion problems, always break the motion into horizontal and vertical components and solve for each separately.

Learn how to solve a projectile motion problem where a soccer ball is kicked at 17.5 m/s at an angle of 30° above the horizontal. Find the time of flight, maximum height, and range using trigonometry and kinematic equations. Suitable for high school students studying physics or mathematics.

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