A man kicks a soccer ball at an angle of 35 degress above the ground and it hits the ground 3.2 seconds later. How high and how far does the ball travel?

To find the height and horizontal distance traveled by the soccer ball, we need to use the equations of projectile motion. These equations assume no air resistance and that the only force acting on the ball is gravity.

Let's break down the problem and solve it step by step:

1. Split the initial velocity into horizontal and vertical components:
- Vertical component (Vsy): Vsy = V * sin(θ)
- Horizontal component (Vsx): Vsx = V * cos(θ)

In this case, the initial velocity is not given, but since we only need the ratios of the components, we can assume an arbitrary value. Let's assume V = 10 meters per second.

Vsy = 10 * sin(35°) ≈ 5.74 m/s
Vsx = 10 * cos(35°) ≈ 8.12 m/s

2. Calculate the time the ball spends in the air:
The time it takes for the ball to hit the ground is given as 3.2 seconds.

3. Calculate the vertical displacement (height):
Using the equation for vertical displacement:
Y = Vsy * t - (1/2) * g * t^2
where g is the acceleration due to gravity (9.8 m/s^2).

Plugging in the values:
Y = 5.74 * 3.2 - (1/2) * 9.8 * 3.2^2 ≈ 5.74 * 3.2 - 1/2 * 9.8 * 10.24 ≈ 18.37 - 50.31 ≈ -31.94 meters

The negative sign indicates that the ball falls below the starting point, so it traveled downward and below the ground level.

To find the maximum height, we can use the fact that the maximum height occurs when the vertical velocity becomes zero. Thus, Vsy = 0 when the ball reaches its highest point.

0 = 5.74 - 9.8 * t
t ≈ 5.74 / 9.8 ≈ 0.5867 seconds

Using this time value, we can calculate the maximum height using the equation for vertical displacement: Y = Vsy * t - (1/2) * g * t^2
Y = 5.74 * 0.5867 - (1/2) * 9.8 * 0.5867^2 ≈ 1.68 meters

Therefore, the ball's maximum height is approximately 1.68 meters.

4. Calculate the horizontal displacement (distance traveled):
Using the equation for horizontal displacement:
X = Vsx * t

Plugging in the values:
X = 8.12 * 3.2 ≈ 25.98 meters

Therefore, the horizontal distance traveled by the ball is approximately 25.98 meters.

To summarize:
- The ball reaches a maximum height of approximately 1.68 meters above its starting point.
- The ball travels horizontally for approximately 25.98 meters.