An object that is 27 cm in front of a convex mirror has an image located 18 cm behind the mirror. How far behind the mirror is the image located when the object is 19 cm in front of the mirror?
Compute the focal length (f) from the object and image information provided.
Then use that f (which will be negative for a convex mirror) to compute the new image position when the object is moved.
1/27 - 1/18 = 1/f
2/54 - 3/54 = -1/54 = 1/f
f = -54 cm
1/19 + 1/Di = -1/54
1/Di = -1/54 - 1/19 = -0.007115
= -1/14.05
Di = -14.05 cm
which means 14.05 cm behind the mirror
An object that is 29 cm in front of a convex mirror has an image located 13 cm behind the mirror. How far behind the mirror is the image located wen the object is 15 cm in front of the mirror?
To solve this problem, you can use the mirror equation for convex mirrors, which is:
1/f = 1/d_o + 1/d_i
Where:
f is the focal length of the mirror,
d_o is the distance of the object from the mirror, and
d_i is the distance of the image from the mirror.
First, let's find the focal length of the mirror. The focal length of a convex mirror is always positive and is equal to half the radius of curvature (R) of the mirror. Since the question doesn't provide the radius of curvature, we will assume a value.
Let's assume the radius of curvature (R) of the convex mirror is -36 cm (negative because it is a convex mirror).
From the formula, we know that 1/f = 1/d_o + 1/d_i.
Substituting the values, we have:
1/(-36) = 1/27 + 1/(-18).
Simplifying this equation, we get:
-1/36 = 1/27 - 1/18.
Now, let's find the value of 1/f. We can multiply both sides of the equation by (-36) to isolate 1/f:
(-1/36)(-36) = 1/27(-36) - 1/18(-36).
This simplifies to:
1/f = -1 + 2.
So, 1/f = 1.
Therefore, the focal length of the mirror (f) is -36 cm.
Now, let's use the mirror equation to find the distance of the image (d_i) when the object is 19 cm in front of the mirror.
We have:
1/f = 1/d_o + 1/d_i.
Substituting the values, we get:
1/(-36) = 1/19 + 1/d_i.
Simplifying this equation, we have:
-1/36 = (1/19) + (1/d_i).
To find d_i, we can subtract 1/19 from both sides of the equation:
-1/36 - (1/19) = (1/d_i).
Now, let's simplify the equation further:
-3/108 - 6/108 = (1/d_i).
Combining the fractions on the left side, we get:
-9/108 = (1/d_i).
To isolate d_i, we can take the reciprocal of both sides of the equation:
d_i = 108/(-9).
Simplifying this expression, we find that:
d_i = -12 cm.
Therefore, when the object is 19 cm in front of the mirror, the image is located 12 cm behind the mirror.