Problem 72 - Fundamentals of Physics

Question

A rugby player runs with the ball directly toward his opponent's goal, along the positive direction of an \(x\) axis. He can legally pass the ball to a teammate as long as the ball's velocity relative to the field does not have a positive \(x\) component. Suppose the player runs at speed \(4.0 \mathrm{~m} / \mathrm{s}\) relative to the field while he passes the ball with velocity \(\vec{v}_{B P}\) relative to himself. If \(\vec{v}_{B P}\) has magnitude \(6.0 \mathrm{~m} / \mathrm{s}\), what is the smallest angle it can have for the pass to be legal?

Step-by-Step Solution

Answer

The smallest angle for a legal pass is approximately \(131.8^\circ\).

Step 1: Understanding Relative Velocity

The problem involves two reference frames: the rugby player's reference frame and the field's reference frame. The ball's velocity relative to the field is given by \( \vec{v}_{B} = \vec{v}_{BP} + \vec{v}_{P} \), where \( \vec{v}_{BP} \) is the ball's velocity relative to the player and \( \vec{v}_{P} \) is the player's velocity relative to the field.

Step 2: Define the Velocity of the Ball

The velocity of the player, \( \vec{v}_{P} \), is given as \(4.0 \mathrm{~m/s}\) in the positive \(x\) direction. The ball’s velocity relative to the player, \( \vec{v}_{BP} \), has a magnitude of \(6.0 \mathrm{~m/s}\) but an unknown direction. We express this as \( \vec{v}_{BP} = v_{BP,x} \hat{i} + v_{BP,y} \hat{j} \). Since the direction is unknown, we use trigonometry: \( v_{BP,x} = 6.0 \cos(\theta) \) and \( v_{BP,y} = 6.0 \sin(\theta) \).

Step 3: Calculate the Ball's Velocity Relative to the Field

We substitute the components into the relative velocity equation: \( \vec{v}_{B} = (6.0 \cos(\theta) + 4.0) \hat{i} + 6.0 \sin(\theta) \hat{j} \).

Step 4: Determine the Condition for a Legal Pass

For the pass to be legal, the \(x\)-component of \( \vec{v}_{B} \) must not be positive, meaning \( 6.0 \cos(\theta) + 4.0 \leq 0 \).

Step 5: Solve the Inequality

Solve \( 6.0 \cos(\theta) + 4.0 \leq 0 \) to find \( \cos(\theta) \leq -\frac{4.0}{6.0} \), which simplifies to \( \cos(\theta) \leq -\frac{2}{3} \).

Step 6: Find the Smallest Angle

The smallest angle \( \theta \) in the first two quadrants such that \( \cos(\theta) = -\frac{2}{3} \) is \( \theta = \cos^{-1}(-\frac{2}{3}) \). Calculating this gives \( \theta \approx 131.8^\circ \).

Key Concepts

Rugby

Rugby is a fast-paced and strategic team sport, popular worldwide, especially in countries like New Zealand, Australia, and England. The game involves two teams competing to score the most points, primarily by carrying or kicking an oval-shaped ball into the opposition's territory. A key aspect of rugby is the precise passing between players to navigate the ball across the field.

Passing in rugby must adhere to certain rules, maintaining the integrity and competitiveness of the game. This involves understanding physics concepts like relative velocity to ensure each pass complies with the game's regulations. Players need to master these techniques to excel and win matches. Successful play requires a blend of strategy, teamwork, and physical prowess.

Legal Pass

In rugby, a legal pass is fundamental, ensuring the game's flow and maintaining fair play. A legal pass means that the ball should not have a forward component relative to the field. This means when the ball is passed, it should not move forward in relation to the goal line of the field.

To achieve this, players must throw the ball at an angle and speed that ensures the ball's forward momentum is neutralized or reversed to conform to the rules. In our example, the rugby player must carefully consider his velocity and the angle of the pass to ensure compliance with this rule.

• Understanding the non-forward motion of a legal pass prevents penalties during play.
• Mastering the art of passing contributes significantly to effective team strategy.

Recognizing the physics behind the ball's movement helps players make quick decisions and successful plays.

Velocity Components

Velocity components are crucial in determining the direction and magnitude of a pass in rugby, especially when ensuring it's a legal pass. The vector representation of velocity includes both magnitude and direction.

When analyzing the ball's velocity, it's important to break it down into two components:

• The horizontal component (\(v_{BP,x}\))
• The vertical component (\(v_{BP,y}\))

These are determined using trigonometric functions such as cosine and sine with respect to the angle at which the ball is thrown.

For instance, we define these components as:

• \( v_{BP,x} = 6.0 \cos(\theta) \)
• \( v_{BP,y} = 6.0 \sin(\theta) \)

This breakdown allows players to analyze the motion accurately and adjust the angle to meet game regulations. Mastery of these components is essential for executing strategic and compliant passes.

Vector Analysis

Vector analysis is a mathematical tool used to analyze the movement and forces acting on an object, vital for understanding passes in rugby. By breaking down velocities into components, and analyzing these vectors, players and coaches can predict and control the ball's trajectory.

In the context of rugby, a player uses vector analysis to ensure a pass remains non-forward relative to the field. You have:

• The velocity vector of the ball relative to the player (\( \vec{v}_{BP} \))
• The velocity vector of the player relative to the field (\( \vec{v}_{P} \))

By combining these vectors, we derive the ball's velocity vector relative to the field, \( \vec{v}_{B} = (6.0 \cos(\theta) + 4.0) \hat{i} + 6.0 \sin(\theta) \hat{j} \).

Understanding this equation helps in comprehending how different angles and player speeds impact the legality of a pass, crucial for strategy and compliance on the field.