Calculating the Horizontal Distance Traveled by a Kicked Football: The Mathematical Approach

Understanding the motion of a football kicked at an angle involves the application of fundamental physics principles. In this article, we will explore how to calculate the horizontal distance traveled by a football given an initial speed and angle of projection. We will break down the solution into several steps and derive the necessary formulas to achieve an accurate result.

Basic Concepts and Steps

When a football is kicked, the initial velocity can be decomposed into its horizontal and vertical components. This allows us to analyze the motion in two dimensions separately. The key steps are as follows:

Decompose the initial velocity. Calculate the time of flight. Calculate the horizontal distance.

Decomposing the Initial Velocity

Let's start with an example where a football is kicked with an initial speed of 10 m/s at an angle of 50°.

Step 1: Decompose the Initial Velocity

The initial speed is given by:

[$$v_0 10 , text{m/s}$$]

The angle of projection is:

[$$theta 50^circ$$]

The horizontal and vertical components of the initial velocity are calculated using trigonometric functions:

[$$v_x v_0 cdot cos(theta) 10 cdot cos(50^circ) approx 6.43 , text{m/s}$$] [$$v_{0y} v_0 cdot sin(theta) 10 cdot sin(50^circ) approx 7.66 , text{m/s}$$]

Here, (cos(50^circ) approx 0.6428) and (sin(50^circ) approx 0.7660).

Calculating the Time of Flight

The time of flight of a projectile launched and landing at the same height can be calculated using:

[$$T frac{2v_{0y}}{g}$$]

Where (g) is the acceleration due to gravity, approximately 9.81 m/s2.

Step 2: Calculate the Time of Flight

Substituting the values:

[$$T frac{2 cdot 7.660}{9.81} approx 1.56 , text{s}$$]

This is the total time the football remains in the air.

Calculating the Horizontal Distance

The horizontal distance is given by:

[$$R v_x cdot T$$]

Step 3: Calculate the Horizontal Distance

Substituting the values:

[$$R approx 6.43 cdot 1.56 approx 10.03 , text{m}$$]

Thus, the horizontal distance traveled by the football is approximately 10.03 meters.

Deriving the Range Formula

For a more theoretical approach, let’s derive the range formula for projectile motion.

Range Using the Range Formula

The range (R) for a projectile can be expressed as:

[$$R frac{2{v_0}^2 sin theta cos theta}{g} frac{{v_0}^2 sin 2theta}{g}$$]

Using the trigonometric identity (2 sin theta cos theta sin 2theta), we can simplify the formula to:

[$$R frac{{v_0}^2 sin 2theta}{g}$$]

Step 4: Applying the Range Formula

Substituting (v_0 10 , text{m/s}), (theta 50^circ), and (g 9.81 , text{m/s}^2):

[$$R frac{7^2 sin 100^circ}{10} frac{49 sin 100^circ}{10} approx 4.82 , text{m}$$]

Rounding to one significant digit, the horizontal distance is approximately 5 meters.

Conclusion

Both methods confirm that the horizontal distance traveled by the football is approximately 10.03 meters using the manual decomposition method, and 5 meters using the range formula. These calculations offer a deeper understanding of projectile motion and the factors influencing the trajectory of a kicked football.