Light Beam Speed on Shoreline from P - 16km

• MHB
• joy2
In summary, the light beam of a lighthouse located 16 km off-shore makes 5 revolutions per minute, which is equivalent to $\dfrac{\pi}{6}$ rad/sec. By considering a right triangle formed by the light beam, shoreline, and perpendicular distance, with $\theta$ as the angle between the light beam and perpendicular distance, and $x$ as the distance from the light beam to the shoreline, we can write a trig equation that relates $x$, $\theta$, and the fixed 16 km distance. Taking the time derivative of this equation, we can determine the speed of the light beam along the shoreline when $x=3$ km. It is important to keep track of units throughout the calculation.
joy2
A lighthouse is located on a small island 16 km off-shore from the nearest point P on a straight shoreline. Its light makes 5 revolutions per minute. How fast is the light beam moving along the shoreline when it is shining on a point 3 km along the shoreline from P?

$\dfrac{d\theta}{dt}$ = 5 rpm = $\dfrac{10\pi}{60 \, sec} = \dfrac{\pi}{6}$ rad/sec

consider the right triangle formed by the light beam, the shoreline, and the perpendicular distance from the light house to the shoreline (recommend you make a sketch)

let $\theta$ be the angle between the light beam and the perpendicular distance, and $x$ be the distance from where the light beam intersects the shoreline to where the perpendicular distance segment intersects the shoreline. You are given the fixed perpendicular distance.

Using the aforementioned right triangle, write a trig equation that relates $x$, $\theta$, and the 16 km distance.

Take the time derivative of the equation and determine $\dfrac{dx}{dt}$ when $x=3$ km.

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1. What is the speed of a light beam on a shoreline from P to 16km?

The speed of light on a shoreline from point P to 16km is approximately 299,792,458 meters per second. This is the speed of light in a vacuum, which is the fastest possible speed for any object in the universe.

2. How is the speed of light determined on a shoreline?

The speed of light on a shoreline is determined using a technique called time-of-flight measurement. This involves measuring the time it takes for a light beam to travel from point P to 16km and back, and then using this time to calculate the speed of light.

3. Does the speed of light change on a shoreline?

No, the speed of light does not change on a shoreline. It is a constant value that is the same everywhere in the universe. However, the speed of light can be affected by the medium it is traveling through, such as air or water.

4. How does the speed of light on a shoreline compare to the speed of light in other environments?

The speed of light on a shoreline is the same as the speed of light in a vacuum, which is the fastest possible speed for any object in the universe. However, the speed of light can be slower when traveling through a medium such as air or water.

5. Why is it important to know the speed of light on a shoreline?

Knowing the speed of light on a shoreline is important for various scientific and technological applications. This includes understanding the behavior of light in different environments, developing communication systems, and conducting accurate measurements and experiments.

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