Find the location of the diver's reflection in the convex mirror

[hraghav]

Homework Statement

A SCUBA diver is practicing in a calm swimming pool. The swimming pool has a large, convex mirror above it. The distance between the top of the water and the surface of the mirror is
193cm. The diver is a distance of 181.2cm beneath the surface of the pool. The mirror has a radius of curvature of 222.4cm. The water has an index of refraction of
n=1.54. The index of refraction of air is n=1. This is illustrated in the image below. What is the location of the diver's reflection in the mirror, relative to the surface of the water (positive is above the water, negative is below), as would be seen by someone located above the water, looking into the mirror?

Relevant Equations

Dapparent = Dreal / n
f = -R / 2
1 / f = 1 / do + 1/di

Dapparent = Dreal / n
Dapparent = 181.2 / 1.54 = 117.66 cm

the total distance from the mirror to the diver is: do = 193 cm + 117.66 cm = 310.66 cm
As the object is in front of the convex mirror, do = -310.66 cm

f = -R/2 = -222.4 / 2 = -111.2 cm
1 / f = 1 / do + 1/di
1 / -111.2 = 1 / -310.66 + 1/di
di = -173.1938 cm

location is 193 cm -173.1938 cm = 19.80 cm

But this is not correct. Could someone please help me spot my mistake here.

Thanks!

 

[kuruman]

Question: How would your calculation differ if there were no water in the pool?
Answer: It would not. Does this make sense to you?

 

[hraghav]

Sorry I didn't quite understand what you mean

 

[hraghav]

Question: How would your calculation differ if there were no water in the pool?
Answer: It would not. Does this make sense to you?

I took the total distance from the mirror to the diver as: do = 193 cm + 181.2 cm = 374.2 cm
So
1 / f = 1 / do + 1/di
1 / -111.2 = 1 / -374.2 + 1/di
di = -158.2168 cm

location is 193 cm -158.2168 cm = 34.7831 cm

But this is still not correct. Is this what you had meant or did I interpret it incorrectly?

 

[kuruman]

You have not taken into account the bending of the light rays as they enter the pool. See the discussion here
https://www.physicsforums.com/threa...pth-equation-explained-and-simplified.872987/ You need to find the distance between the mirror and the diver as the diver sees it.

 

[hraghav]

You have not taken into account the bending of the light rays as they enter the pool. See the discussion here
https://www.physicsforums.com/threa...pth-equation-explained-and-simplified.872987/ You need to find the distance between the mirror and the diver as the diver sees it.

From my very first attempt I got my location as 193 cm -173.1938 cm = 19.80 cm
to account for the bending light I tried to do 19.80 / 1.54 = 12.857 but this is still not correct. I am very confused at the moment because I am not sure how to use the refractive index in finding the location. Could you please give me some hints as that would be very helpful in understanding what is going wrong here.
Thank you

 

[haruspex]

You need to find the distance between the mirror and the diver as the diver sees it.

I would put it differently: find the distance from the diver to the mirror as the mirror sees it.

 

[hraghav]

I would put it differently: find the distance from the diver to the mirror as the mirror sees it.

Would that happen to be 193 -181.2 = 11.8 cm?

 

[haruspex]

Would that happen to be 193 -181.2 = 11.8 cm?

Now you are just making wild guesses.
If you were suspended 193cm above the pool, where would the diver appear to be?

 

[hraghav]

Now you are just making wild guesses.
If you were suspended 193cm above the pool, where would the diver appear to be?

directly beneath me so would that be 310.66 cm as it accounts for the refractive index?

 

[haruspex]

directly beneath me so would that be 310.66 cm as it accounts for the refractive index?

Yes. So what's next?

 

[hraghav]

Yes. So what's next?

Finding di using the lens equation which I got as -173.194cm?

 

[haruspex]

Finding di using the lens equation which I got as -173.194cm?

Sorry, I just realised you did all this in post #1. I was misled by the ensuing posts and failed to check back all the way.

So the only things I see that could be wrong are the signs in your mirror formula.
At https://www.physicsclassroom.com/class/refln/Lesson-4/The-Mirror-Equation-Convex-Mirrors the formula is 1/di+1/do=1/f where do is (necessarily) the distance in front of the mirror, f is positive, and di is behind the mirror.

 

[hraghav]

Sorry, I just realised you did all this in post #1. I was misled by the ensuing posts and failed to check back all the way.

So the only things I see that could be wrong are the signs in your mirror formula.
At https://www.physicsclassroom.com/class/refln/Lesson-4/The-Mirror-Equation-Convex-Mirrors the formula is 1/di+1/do=1/f where do is (necessarily) the distance in front of the mirror, f is positive, and di is behind the mirror.

But I am getting -19.8054 as my answer and this is incorrect, thats the reason why I am confused as I am not sure why it is wrong. Is my final answer correct?

 

[haruspex]

But I am getting -19.8054 as my answer and this is incorrect, thats the reason why I am confused as I am not sure why it is wrong. Is my final answer correct?

I say again

di is behind the mirror

The image in a convex mirror is never in front of it.

 

[hraghav]

I say again

The image in a convex mirror is never in front of it.

I am sorry but does this mean I add 193 to 173.194 or am I thinking in the wrong direction?

 

[haruspex]

I am sorry but does this mean I add 193 to 173.194 or am I thinking in the wrong direction?

Yes.
But you are keeping too many sig figs. The numbers you are given are a mix of three and four sig figs, so certainly do not quote more than four in the answer,

 

[hraghav]

Yes.
But you are keeping too many sig figs. The numbers you are given are a mix of three and four sig figs, so certainly do not quote more than four in the answer,

But my answer is still not correct, I am getting 366.19 which is wrong

 

[haruspex]

But my answer is still not correct, I am getting 366.19 which is wrong

It just dawned on me that the info at the link I posted cannot be right. If we set f to infinity we should get di and do as equal but on opposite sides, i.e. flat mirror.
Here's what it should say:
"1/f=1/di+1/do where all three are measured as distances in front of the mirror."
So we have f=-111.2, di=310.7, and di will turn out negative, meaning it is |di| above (behind) the mirror.

Note that in all convex mirror cases f will be negative and do positive, so di is guaranteed negative.

 

[hraghav]

It just dawned on me that the info at the link I posted cannot be right. If we set f to infinity we should get di and do as equal but on opposite sides, i.e. flat mirror.
Here's what it should say:
"1/f=1/di+1/do where all three are measured as distances in front of the mirror."
So we have f=-111.2, di=310.7, and di will turn out negative, meaning it is |di| above (behind) the mirror.

Note that in all convex mirror cases f will be negative and do positive, so di is guaranteed negative.

Nevermind I got the right answer, Thanks a lot for your help :)