About the years 214-212 BC, the Greek philosopher and scientist Aρχιμηδηζ is said to have repelled a Roman attack with a "death-ray" concave mirror, using sunlight to set the approaching fleet on fire. Assume an early morning attack from the East, with the Sun directly behind the ships. If the approaching fleet is 400 meters away, what would need to be the radius of curvature of the mirror in order to achieve maximum effect? (in m)
800m
To calculate the radius of curvature of the mirror for maximum effect, we need to consider the geometry and properties of concave mirrors. Let's break down the steps to find the answer:
Step 1: Understand the concept
A concave mirror is a curved mirror where the surface curves inward. It is known that rays of light parallel to its principal axis converge to a point called the focal point. This point is located at a distance equal to half the radius of curvature from the mirror's surface.
Step 2: Determine the required focal length
In this case, we want to set the approaching fleet on fire, so we need to focus as much sunlight as possible on the ships. To maximize the concentration of sunlight, we want the focal point of the mirror to coincide with the location of the ships, which are 400 meters away. Therefore, the focal length of the mirror should be 400 meters.
Step 3: Calculate the radius of curvature
We know that the focal length (f) is half the radius of curvature (R), so we can use the relationship f = R/2 to find the radius of curvature.
f = 400 meters
R = ?
R = 2f
R = 2 * 400
R = 800 meters
Therefore, the radius of curvature of the mirror needs to be 800 meters to achieve maximum effect.
It's important to note that this calculation assumes ideal conditions, perfect alignment, and other factors not accounted for. The story of Archimedes using a "death-ray" is often considered a legend, and no historical evidence supports its veracity.