A ball is kicked with a velocity of 20 m/s at 45°. Calculate the total time of flight, maximum height, and range.

To solve this problem, we can break down the motion of the ball into horizontal and vertical components.

Given:
Initial velocity (v) = 20 m/s
Launch angle (θ) = 45°
Acceleration due to gravity (g) = 9.81 m/s^2

1. Calculate the horizontal and vertical components of the initial velocity:
Horizontal component (v_x) = v * cos(θ)
v_x = 20 * cos(45°) = 20 * 0.7071 ≈ 14.14 m/s

Vertical component (v_y) = v * sin(θ)
v_y = 20 * sin(45°) = 20 * 0.7071 ≈ 14.14 m/s

2. Calculate the time of flight:
The time of flight can be calculated using the vertical component of the motion. The time taken for the ball to reach its maximum height is the same as the time taken for the ball to fall back to the ground.

Time to reach maximum height (t) = v_y / g
t = 14.14 / 9.81 ≈ 1.44 seconds

Total time of flight = 2 * t
Total time of flight = 2 * 1.44 ≈ 2.88 seconds

3. Calculate the maximum height:
The maximum height can be calculated using the vertical component of the motion.

Maximum height (h) = (v_y)^2 / (2 * g)
h = (14.14)^2 / (2 * 9.81) ≈ 10.16 meters

4. Calculate the range:
The range can be calculated using the horizontal component of the motion.

Range = v_x * total time of flight
Range = 14.14 * 2.88 ≈ 40.7 meters

Therefore, the total time of flight is approximately 2.88 seconds, the maximum height is approximately 10.16 meters, and the range is approximately 40.7 meters.

1. Total Time of Flight

The time of flight is the total time the ball spends in the air, from when it is kicked to when it lands. To calculate the time of flight, we can use the following formula:

```
Time of Flight = (2 - Vertical Velocity) / Acceleration Due to Gravity
```

In this case, the vertical velocity is 20 m/s - sin(45°) = 14.14 m/s. The acceleration due to gravity is 9.81 m/s².

Therefore, the time of flight is:

```
Time of Flight = (2 - 14.14 m/s) / 9.81 m/s²
Time of Flight = 2.89 s
```

2. Maximum Height

The maximum height is the highest point the ball reaches in its trajectory. To calculate the maximum height, we can use the following formula:

```
Maximum Height = (Vertical Velocity)² / (2 - Acceleration Due to Gravity)
```

In this case, the vertical velocity is 14.14 m/s. The acceleration due to gravity is 9.81 m/s².

Therefore, the maximum height is:

```
Maximum Height = (14.14 m/s)² / (2 - 9.81 m/s²)
Maximum Height = 10.23 m
```

3. Range

The range is the horizontal distance the ball travels from when it is kicked to when it lands. To calculate the range, we can use the following formula:

```
Range = Horizontal Velocity - Time of Flight
```

In this case, the horizontal velocity is 20 m/s - cos(45°) = 14.14 m/s. The time of flight is 2.89 s.

Therefore, the range is:

```
Range = 14.14 m/s - 2.89 s
Range = 40.79 m
```