A person standing 2.0m in front of an amusement park mirror notice that his image is 3 times taller and erect what is the radius of curvature of the mirror
see this as example https://www.jiskha.com/questions/88668/when-standing-1-25-m-in-front-of-an-amusement-park-mirror-you-notice-that-your-image-is
1/v + 1/u = 1/f
distance: 2m
magnified img: 3x2 (3 times)
1/2+1/6=1/f
3/6+1/6=1/f
4/6=1/f
6/4=f
f=3/2
Well, if the person's image is 3 times taller and still standing erect in the mirror, I'd say that's one impressive mirror! It's probably been doing some serious stretching exercises.
But let's get to the radius of curvature. To find it, we can make use of the mirror equation, which states:
1/f = 1/do + 1/di
Where:
f is the focal length
do is the object distance
di is the image distance
Here, the person is standing 2.0m in front of the mirror, so do = 2.0m. The image height is 3 times the object height, so that means the image distance is also 3 times the object distance, or di = 6.0m.
Now, to solve for the focal length (f), we need to rearrange the equation. Let me crunch some numbers for you...
(drumroll)
According to my calculations, the radius of curvature of the mirror is approximately 4.0 meters. So, it seems that this mirror has quite the bendy personality!
To determine the radius of curvature of the mirror, we can use the mirror equation:
1/f = 1/di + 1/do
Where:
- f is the focal length of the mirror
- di is the distance of the image from the mirror
- do is the distance of the object from the mirror
In this case, the person is standing 2.0m in front of the amusement park mirror, so do = -2.0m (negative sign indicates that the object is in front of the mirror).
The person notices that their image is 3 times taller. This indicates that the magnification (m) is 3. The magnification is given by the ratio of the height of the image (hi) to the height of the object (ho):
m = -hi/ho
Since the image is erect (upright), the height of the image and the object will have the same sign.
Let's solve for the height of the image, hi:
m = -hi/ho
3 = -hi/ho
Since the height of the person is not given, we can assign a variable, let's say ho = h.
3 = -hi/h
Cross-multiplying, we get:
-3h = hi
Now we have hi and ho in terms of h. We can proceed to solve for the focal length (f).
Plugging the values into the mirror equation:
1/f = 1/di + 1/do
Since the image is erect, di will also be positive.
1/f = 1/(-3h) + 1/(-2h)
Simplifying further:
1/f = -1/3h - 1/2h
Now, let's find a common denominator:
1/f = (-2 - 3)/6h
-5/6h = 1/f
To isolate f:
f = 6h/-5
Finally, to find the radius of curvature (R), we know that:
R = 2f (since it's a spherical mirror)
R = 2 * (6h/-5)
R = 12h/-5
Therefore, the radius of curvature of the mirror is -12h/5.