Hockey Scoring Angles: Position and Distance Effects

Answer

Lars has a worse chance to score when positioned directly in front of a post because the shooting angle decreases compared to being centered. Moving closer to the net increases the shooting angle (better chance), while moving farther away decreases it (worse chance).

Observation of Figure

The diagram illustrates a hockey puck positioned $8\text{ m}$ away from a goal net. The puck is aligned "directly in front of one of the posts," which creates a right-angled triangle between the puck, the near post, and the far post. The goal width is labeled as $B$ (Standard hockey goals are $1.83\text{ m}$ wide).

Explanation

1. Defining the "Chance to Score" Geometric Model In sports physics, the "chance to score" is mathematically modeled by the shooting angle ($\theta$). A larger angle provides a wider target area for the puck to enter the net. We assume Lars's accuracy is constant and focus strictly on the geometric availability of the goal. $\theta = \arctan\left(\frac{\text{Opposite}}{\text{Adjacent}}\right)$ This formula calculates the angular width of the goal from Lars's perspective.

2. Analyzing the Shift to the Post (Part A) When centered, Lars forms an isosceles triangle with the posts. By moving directly in front of one post, he creates a right triangle. Despite the distance to the near post remaining roughly similar, the effective angular width $\theta$ decreases because the "average" distance to the goal plane increases and the perspective becomes skewed. ⚠️ This step is required on exams: To prove a change in "chance," you must explicitly state that the angular width $\theta$ is the dependent variable.

3. Analyzing Distance Changes (Part B) As Lars moves directly closer to the net (decreasing the "Adjacent" side of our triangle), the denominator in the tangent function decreases. $\text{As } d \to 0, \theta \to 90^\circ \text{ (or } 180^\circ \text{ if centered)}$ Conversely, as he moves farther away ($d \to \infty$), the angle $\theta$ approaches $0^\circ$.

Movement	Effect on Angle ($\theta$)	Chance to Score
Move in front of post	Decreases	Worse
Move closer to net	Increases	Better
Move farther from net	Decreases	Worse

Final Answer

Lars has a worse chance in part (a) because the angular width of the goal is maximized when a player is centered. In part (b), moving closer creates a better chance (larger angle), while moving farther creates a worse chance (smaller angle). $\boxed{\text{Larger Angle} = \text{Better Chance}}$

Common Mistakes

• Distance vs. Angle Confusion: Students often assume that being "closer" to one post compensates for the skew. In geometry, the total angular spread $\theta$ is what defines the target size, and this is always maximized when the observer is on the perpendicular bisector of the goal line (centered).
• Ignoring Dimensions: On exams, ensure you specify that $B$ (width) remains constant while $d$ (distance) varies; otherwise, the ratio is undefined.