After hitting the ball into a water trap, a golfer looks into the pond and spies the ball within apparent easy reach. Reaching in to retrieve the ball, the golfer is surprised to find that it cannot be grasped even with a fully extended arm. Explain why the golfer was deceived into thinking that the location of the ball was close at hand.

Short Answer

Expert verified
Answer: The difficulty in reaching the ball underwater is due to the refraction of light. When the golfer looks at the ball, the light rays from the ball pass through two different media (water and air), causing the light to bend (refract) and subsequently altering the ball's perceived position. According to Snell's Law, the light bends away from the normal when exiting the water, resulting in the angle of refraction being larger than the angle of incidence. The golfer's brain interprets these refracted light rays as having traveled in a straight line, causing them to perceive the ball to be closer to the surface than it actually is. This misinterpretation makes it challenging for the golfer to grasp the ball, even with a fully extended arm.

Step by step solution

01

Understanding Light Refraction

When light passes from one medium (e.g. water) to another medium (e.g. air), it changes its speed as each medium has a different index of refraction. This causes the light rays to bend (or refract), altering the path of the light and, consequently, the perceived position of the object.
02

Introducing Snell's Law

Snell's Law is a formula that describes the relationship between the angle of incidence (\(\theta_{1}\)), the angle of refraction (\(\theta_{2}\)) and the indices of refraction of both media (n1 and n2). The formula for Snell's Law is given by: \[n_{1}\times \sin(\theta_{1}) = n_{2}\times \sin(\theta_{2})\]
03

Exploring the Relationship between Air, Water, and Light

When the golfer looks at the ball in the water, the light rays from the ball pass through two different media: water and air. The light rays bend due to the change in speed and refractive index as they pass from water (with a refractive index (\(n_{2}\)) of about 1.33) to air (with a refractive index (\(n_{1}\)) of about 1).
04

Analyzing the Angles of Incidence and Refraction

Let's assume the angle of incidence when the light ray emerges from the water is \(\theta_{1}\) and the angle of refraction in air is \(\theta_{2}\). Because the refractive index of water is greater than that of air, according to Snell’s Law, the light will bend away from the normal (the imaginary line that is perpendicular to the water surface) when exiting the water, causing the angle of refraction (\(\theta_{2}\)) to be larger than the angle of incidence (\(\theta_{1}\)).
05

Understanding the Golfer's Perception

Due to the refraction of light, the golfer perceives the ball to be closer to the surface than it actually is. The brain interprets the refracted light rays as having traveled in a straight line, causing the golfer to perceive the object (the golf ball) to be in a higher position than its actual location. This explains why the golfer is unable to grasp the ball even with a fully extended arm.

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