Impact Force and Surface Area Relationship in Sports

When a player’s foot contacts the ground while playing a sport, the force exerted on the ground (impact force) is distributed over the contact area of the foot with the pitch. We aim to derive a relationship between the impact force, \( F \), and the surface area of contact, \( A \).

Problem Formulation

Given:

• Impact force on the ground, \( F \) (measured in newtons).
• Surface area of the foot in contact with the ground, \( A \) (measured in square meters).
• Pressure exerted on the ground due to the impact force, \( P \).

We want to derive a formula that expresses \( A \) in terms of \( F \) and \( P \).

Solution

Pressure \( P \) is defined as the force applied per unit area. This can be expressed as:

Rearranging this equation to solve for \( A \) in terms of \( F \) and \( P \):

Therefore, the surface area of contact \( A \) is inversely proportional to the pressure \( P \) for a given impact force \( F \). As the pressure increases, the contact area decreases, and vice versa.

Graph of Area vs Impact Force

For a constant pressure \( P \), the relationship between area \( A \) and force \( F \) is linear:

Below is a simple plot of area \( A \) versus force \( F \) for a constant pressure \( P \):

Conclusion

The relationship between area \( A \) and impact force \( F \) at constant pressure \( P \) is linear, as derived. This relationship suggests that as force increases, the contact area required to maintain the same pressure increases proportionally.