To solve this problem, we can treat it as a projectile motion question where a football is kicked at an initial velocity $v_0 = 20.0 \, \text{m/s}$ and at an angle $\theta = 37.0^\circ$ to the horizontal. We are tasked with finding two things:

1. The time the ball takes to hit the ground (i.e., total flight time).
2. The horizontal distance the ball travels (i.e., the range).

Step 1: Break the Initial Velocity into Components

We need to resolve the initial velocity into its horizontal and vertical components using trigonometry:

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

Where:

• $v_0 = 20.0 \, \text{m/s}$
• $\theta = 37.0^\circ$

Calculating the components: $v_{0x} = 20.0 \, \text{m/s} \times \cos(37^\circ) = 20.0 \times 0.7986 = 15.97 \, \text{m/s}$ $v_{0y} = 20.0 \, \text{m/s} \times \sin(37^\circ) = 20.0 \times 0.6018 = 12.04 \, \text{m/s}$

Step 2: Time of Flight

The total time of flight can be determined by analyzing the vertical motion. The ball will rise to a maximum height and then fall back to the ground. The equation for vertical displacement is:

$y = v_{0y} t - \frac{1}{2} g t^2$

Since the ball starts and ends at ground level, $y = 0$. Thus, the equation becomes:

$0 = v_{0y} t - \frac{1}{2} g t^2$

Factoring out $t$ gives:

$t(v_{0y} - \frac{1}{2} g t) = 0$

This equation has two solutions:

1. $t = 0$ (the time when the ball is kicked).
2. $t = \frac{2v_{0y}}{g}$ (the total time of flight).

Substituting the values for $v_{0y}$ and $g = 9.81 \, \text{m/s}^2$:

$t = \frac{2 \times 12.04 \, \text{m/s}}{9.81 \, \text{m/s}^2} = \frac{24.08}{9.81} = 2.45 \, \text{seconds}$

Step 3: Horizontal Range

The horizontal range (distance) can be found using the horizontal velocity and the total time of flight:

$\text{Range} = v_{0x} \times t$

Substituting the values:

$\text{Range} = 15.97 \, \text{m/s} \times 2.45 \, \text{seconds} = 39.15 \, \text{m}$

Final Answers:

1. Time of flight: 2.45 seconds
2. Range (horizontal distance): 39.15 meters

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the maximum height the ball reaches?
2. How would the time of flight change if the ball were kicked at a different angle?
3. What happens to the horizontal range if the initial velocity is increased to 25 m/s?
4. Can air resistance affect the time of flight and range in a real-world scenario?
5. How would the problem change if the ball were kicked from an elevated platform?

Tip: In projectile motion, the horizontal velocity remains constant if air resistance is neglected, while the vertical velocity changes due to gravity.