Chapter 36, Problem 36.27P

You unconsciously estimate the distance to an object from the angle it subtends in your field of view. This angle θ in radians is related to the linear height of the object h and to the distance d by θ = h/d. Assume yon are driving a car and another car, 1.50 m high, is 24.0 m behind you. (a) Suppose your car has a flat passenger-side rearview mirror, 1.55 m from your eyes. How far from your eyes is the image of the car following you? (b) What angle does the image subtend in your field of view? (c) What If? Now suppose your car has a convex rearview mirror with a radius of curvature of magnitude 2.00 m (as suggested in Fig. 35.15). How far from your eyes is the image of the car behind you? (d) What angle does the image subtend at your eyes? (e) Based on its angular size, how far away does the following car appear to be?

To determine

The distance of the image of car that is following the person.

Answer to Problem 36.27P

The image of the car is at $25.6\text{m}$ form the observer’s eye.

Explanation of Solution

Given info: The height of the following car is $1.50\text{m}$. The distance between the observer’s eyes and the flat rear view mirror is $1.55\text{m}$ and the distance between the cars is $24.0\text{m}$

The rear view mirror is flat plane mirror and for the case of plane mirror the object distance is same as the image distance. Therefore the image distance for the following car is same as the car itself.

Hence, the distance of the car from observer eyes as seen through the flat rear view side mirror is,

$d_{\text{ec}}=d_{\text{om}}+d_{\text{image}}$ (1)

Here,

$d_{\text{ec}}$ is the distance of the image of the car from observers eye.

$d_{\text{om}}$ is the distance between the observer’s eye and the rear view mirror.

$d_{\text{image}}$ is the distance of the image from the mirror.

Substitute $1.55\text{m}$ for $d_{\text{om}}$ and $d_{\text{image}}$ in equation (1),

$\begin{array}{l}{d}_{\text{ec}}=d_{\text{om}}+d_{\text{image}}\\ =1.55\text{m+24}\text{m}\\ \text{=25}\text{.6}\text{m}\end{array}$

Conclusion:

Therefore, the image of the car form the observer’s eye is at distance of $25.6\text{m}$.

To determine

The angle subtended by the image for the observer.

Answer to Problem 36.27P

The angle subtended is $0.0587\text{radian}$.

Explanation of Solution

Given info: The height of the following car is $1.50\text{m}$. The distance between the observer’s eyes and the flat rear view mirror is $1.55\text{m}$ and the distance between the cars is $24.0\text{m}$.

Formula to calculate the angle subtended by the object,

$\theta =\frac{h}{d}$ (2)

Here,

$\theta$ is the angle subtended.

$h$ is the height of the object.

$d$ is the distance of the object from the eye.

For the case of plane mirrors the object height and image height are equal and object distance and image distance are equal. Hence, to find the angle subtended by the image of the following car

Substitute $1.50\text{m}$ for $h$ and $25.55\text{m}$ for $d$ in the equation (2).

$\begin{array}{c}\theta =\frac{h}{d}\\ \theta =\frac{1.50\text{m}}{25.50\text{m}}\\ =0.0587\text{radians}\end{array}$

Conclusion:

Therefore, the angle subtended is $0.0587\text{radians}$.

To determine

The image distance from the observers eye if the rear view mirror is a convex mirror.

Answer to Problem 36.27P

The image is at $\text{2}\text{.51}\text{m}$ from the observer’s eyes.

Explanation of Solution

Given info: The height of the following car is $1.50\text{m}$. The distance between the observer’s eyes and the convex rear view mirror is $1.55\text{m}$ and the distance between the cars is $24.0\text{m}$. The radius of curvature of the mirror is $2.00\text{m}$.

Formula to calculate the image distance form a convex mirror for a given object is

$\frac{1}{p}+\frac{1}{q}=\frac{2}{R}$ (3)

Here,

$p$ is the object distance.

$q$ is the image distance.

$R$ is the radius of the curvature.

Substitute $24\text{m}$ for $p$, $-2\text{m}$ for $R$ in equation (1),

Negative $R$ because it is a convex lens.

$\begin{array}{l}\frac{1}{p}+\frac{1}{q}=\frac{2}{R}\\ \frac{1}{24\text{m}}+\frac{1}{q}=\frac{2}{-2}\\ q=-0.96\text{m}\end{array}$ (4)

The image distance is negative because the image is formed behind the mirror. Therefore the image distance from the observer’s eye is,

$d=d_{\text{om}}+q$ (5)

Here,

$d$ is the required distance between the observers eye and the image.

$d_{\text{om}}$ is the distance between the observer’s eye and the convex rear view mirror.

Substitute $0.96\text{m}$ for $q$ and $1.55\text{m}$ for $d_{\text{om}}$ in equation (5),

$\begin{array}{c}d=d_{\text{om}}+q\\ d=1.55\text{m+0}\text{.960}\text{m}\\ \text{d=2}\text{.51}\text{m}\end{array}$ (6)

Conclusion:

Therefore, the image of the following car is at $\text{2}\text{.51}\text{m}$ form the observer’s eye in case of the convex rear view mirror.

To determine

The angle subtended in the image of view for the observer in the case of convex rear view mirror.

Answer to Problem 36.27P

The angle subtended is $0.0239\text{radian}$.

Explanation of Solution

Given info: The height of the following car is $1.50\text{m}$. The distance between the observer’s eyes and the flat rear view mirror is $1.55\text{m}$ and the distance between the cars is $24.0\text{m}$. The radius of curvature of the mirror is $2.00\text{m}$.

From Equation (2) formula to calculate the angle subtended by the image,

$\theta =\frac{h^{\prime }}{d}$ (7)

Here,

$\theta$ is the angle subtended.

$h^{\prime }$ is the height of the image.

$d$ is the distance of the image from the observer’s eye.

For the case of convex mirrors the object height and image height are not equal.

Formula to calculate the image height is,

$\frac{h^{\prime }}{h}=-\frac{q}{p}$ (8)

Here,

$h$ is the height of object

Substitute $-0.96\text{m}$ for $q$ form equation (4) $1.50\text{m}$ for $h$ and $24.0\text{m}$ for $p$ in equation (6),

$\begin{array}{c}\frac{h^{\prime }}{h}=-\frac{q}{p}\\ \frac{h^{\prime }}{24\text{m}}=-\left(\frac{-0.96\text{m}}{24\text{m}}\right)\\ h^{\prime }=+0.060\text{m}\end{array}$ (9)

From equation (6) and equation (9) respectively, substitute $+0.060\text{m}$ for $h^{\prime }$ and $\text{2}\text{.51}\text{m}$ for $d$ in equation (8),

$\begin{array}{l}\theta =\frac{h^{\prime }}{d}\\ =\frac{0.060\text{m}}{2.51\text{m}}\\ =0.0239\text{radians}\end{array}$ . (10)

Conclusion:

Therefore, the angle subtended is $0.0239\text{radians}$.

To determine

The distance of the following car based on the angle subtended by the image in the observers eye.

Answer to Problem 36.27P

The distance of the following car based on the angular spread in the human eye is $62.8\text{m}$.

Explanation of Solution

Given info: The height of the following car is $1.50\text{m}$. The distance between the observer’s eyes and the flat rear view mirror is $1.55\text{m}$ and the distance between the cars is $24.0\text{m}$. The radius of curvature of the mirror is $2.00\text{m}$.

Formula to calculate the angle subtended by the image,

${\theta }^{\prime }=\frac{h}{d^{\prime }}$ (11)

Here,

${\theta }^{\prime }$ is the angle subtended by the image as seen by the observer in the mirror.

$h$ is the height of the car.

$d^{\prime }$ is the distance of the car that appears to the observer’s eye.

Substitute $0.0239\text{radians}$ for ${\theta }^{\prime }$ form equation (10) $1.50\text{m}$ for $h$ in equation (11),

$\begin{array}{c}{\theta }^{\prime }=\frac{h}{d^{\prime }}\\ 0.0239\text{radians=}\frac{1.50\text{m}}{d^{\prime }}\\ d^{\prime }=62.8\text{m}\end{array}$

Conclusion:

Therefore, the image appears to be $62.8\text{m}$ when viewed from the convex mirror.