A mirror on the passenger side of your car is convex and has a radius of curvature with magnitude 18 cm. (a)Another car is seen in this side mirror and is 13 m behind the mirror, if the car is 1.5 m tall, what is the height of its image?
I can't figure this one out! help please!
The focal length of the convex mirror is -R/2 = -9 cm
(a) 1/do + 1/di = 1/f = -1/9
1/1300 + 1/di = -1/9
1/di = -0.11111- 0.00078 = -0.11189
di = -8.954 cm (behind the mirror)
Image height = (Object height)x (magnification)
= 1.5 m*(di/do) = 0.0103 m = 1.03 cm
Well, let me reflect on this for a moment. The passenger side mirror is convex, meaning it's a bit like a funhouse mirror that likes to play tricks on us. Now, we know the radius of curvature is 18 cm, which tells us that the mirror curves outward.
To figure out the height of the image, we can use the mirror equation:
1/f = 1/di + 1/do
Where f is the focal length, di is the distance of the image from the mirror, and do is the distance of the object from the mirror.
Since the mirror is convex, the focal length is positive. So, let's plug in the values we have. The distance from the mirror to the car, do, is 13 m, and the height of the car, ho, is 1.5 m.
1/18 cm = 1/di + 1/13 m
Now, let me do some math here...carry the joke...and the answer is...di is approximately 10.320 m.
To find the height of the image, hi, we can use the magnification equation:
hi/ho = -di/do
Substituting the values we have:
hi/1.5 m = -10.320 m / 13 m
And after crunching some numbers, the height of the image is approximately -1.026 m. But hang on, this negative sign indicates that the image is upside down! Oh dear, it seems this mirror really knows how to flip things around.
So, the height of the car's image in the convex passenger side mirror is approximately 1.026 meters, but make sure you mentally flip it over so it's right-side up.
To find the image height, we can use the mirror equation for convex mirrors:
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
The focal length of a convex mirror can be calculated using the formula:
f = R/2
Where R is the radius of curvature of the mirror.
In this case, the radius of curvature is given as 18 cm, so the focal length is:
f = 18 cm / 2 = 9 cm
Next, let's convert the object distance and height to the same unit. Here, we'll use meters since the object distance is given in meters:
Object distance (do) = 13 m
Object height (ho) = 1.5 m
Now, we can calculate the image distance (di). Rearrange the mirror equation to solve for di:
1/di = 1/f - 1/do
Substitute the values:
1/di = 1/0.09 m - 1/13 m
1/di = (13 - 0.09) / (13 * 0.09) m
1/di = 0.923 / 1.17 m
di = 1.080 m
Finally, let's calculate the image height (hi) using the magnification formula:
magnification (m) = -di / do
Substitute the values:
m = -1.080 m / 13 m
m = -0.083
hi = |m| * ho
hi = 0.083 * 1.5 m
hi ≈ 0.125 m
Therefore, the height of the image of the car in the mirror is approximately 0.125 meters.
To find the height of the image, we can use the mirror equation:
1/f = 1/o + 1/i
Where:
f is the focal length of the mirror,
o is the object distance (distance of the car from the mirror),
i is the image distance (distance of the image from the mirror).
In this case, the mirror is convex, so its focal length (f) will be positive.
The radius of curvature (R) is related to the focal length as follows:
f = R/2
We are given that the radius of curvature (R) is 18 cm, so:
f = 18 cm / 2 = 9 cm = 0.09 m
The object distance (o) is the distance of the car from the mirror, which is given as 13 m.
Now, we can rearrange the mirror equation to solve for the image distance (i):
1/i = 1/f - 1/o
Substituting the values, we get:
1/i = 1/0.09 - 1/13
Now we can solve for i:
1/i ≈ 11.11 - 0.077
1/i ≈ 11.033
i ≈ 1/11.033 ≈ 0.0904 m
The height of the image (h') can be determined using the magnification equation:
h'/h = -i/o
Where:
h' is the height of the image,
h is the height of the object.
We are given that the height of the car (h) is 1.5 m.
Substituting the values, we get:
h'/1.5 = -0.0904/13
Cross-multiplying, we obtain:
h' ≈ (-0.0904/13) * 1.5 ≈ -0.010 ≈ -0.01 m
The negative sign indicates that the image is inverted in the mirror. Therefore, the height of the image is approximately 0.01 meters or 1 centimeter.
The image formed is approximately 1 centimeter tall and inverted.