An object has a height of 65 cm and is positioned 6.2 m in front of a spherical mirror. Its image distance is 1.9 m. Calculate the height (including its sign) of the image in centimeters.
To calculate the height of the image, we'll use the mirror equation:
1/f = 1/d_o + 1/d_i
where:
f is the focal length of the mirror
d_o is the object distance
d_i is the image distance
First, we need to calculate the focal length of the mirror using the formula:
f = 1 / (1/d_o + 1/d_i)
Given:
d_o = -6.2 m (negative because the object is in front of the mirror)
d_i = 1.9 m
Substituting the values, we get:
f = 1 / (1/-6.2 + 1/1.9) = 1 / (-0.161 + 0.526) = 1 / 0.365 = 2.74 m
Now that we have the focal length, we can use the magnification equation:
m = -d_i / d_o
Given:
d_o = -6.2 m
d_i = 1.9 m
m = -1.9 / -6.2 = 0.306
The negative sign indicates that the image is inverted.
Finally, we can calculate the height of the image using the magnification equation:
m = h_i / h_o, where h_i is the height of the image and h_o is the height of the object.
Rearranging the equation, we get:
h_i = m * h_o
Given:
h_o = 65 cm = 0.65 m
m = 0.306
Substituting the values, we get:
h_i = 0.306 * 0.65 = 0.199 m
Converting it to centimeters, we get:
h_i = 0.199 * 100 = 19.9 cm
Therefore, the height of the image is 19.9 cm.
To solve this problem, we can use the mirror equation:
1/f = 1/di + 1/do
where:
- f is the focal length of the mirror
- di is the image distance
- do is the object distance
First, let's find the focal length of the mirror using the mirror equation:
1/f = 1/di + 1/do
Since the object distance is given as 6.2 m and the image distance is given as 1.9 m, we can substitute these values into the equation:
1/f = 1/1.9 + 1/6.2
Now, let's solve for f:
1/f = 0.5263 + 0.1613
1/f = 0.6876
f = 1 / 0.6876
f ≈ 1.454 m
Now that we have the focal length, we can use the magnification equation to find the height of the image:
m = -di / do
where m is the magnification. Since the image is virtual (since the object is in front of the mirror), the magnification will be negative.
Substituting the given values into the equation:
m = -1.9 / 6.2
m ≈ -0.3065
Since the magnification is negative, it means that the image is inverted.
Now, we can find the height of the image:
hi / ho = -di / do
where hi is the height of the image and ho is the height of the object.
Substituting the given values into the equation:
hi / 65 = -1.9 / 6.2
hi ≈ (-1.9 / 6.2) * 65
hi ≈ -19.83 cm
The height of the image, including its sign, is approximately -19.83 cm.