Question

Two cars A and B moving on a straight road. The car B passes car A by a relative speed of \[45{\text{ m/s}}\]. At what speed does the driver of car B observe car A in the side mirror of focal length 10 cm when car A is at a distance of 1.9 m from car B?
A. \[\dfrac{9}{{80}}\]
B. \[ - \dfrac{9}{{80}}\]
C. \[ - \dfrac{7}{{80}}\]
D. \[\dfrac{7}{{80}}\]

Answer 1

Hint: In this question, we need to find the speed of car B when its driver observes car A in the side mirror. For this, we need to find the magnification of the mirror. After that, we will find the speed of car B.

Formula used:
The formula for the magnification of a spherical mirror is given by
\[m = \dfrac{f}{{f - u}}\]
Here, \[m\] is the magnification factor, \[f\] is the focal length of a mirror and \[u\] is the object distance.
Also, the speed of a car B is given by
\[{v_B} = - {m^2} \times \left( {{v_A}} \right)\]
Here, \[{v_B}\] is the speed of car B, \[{v_A}\] is the speed of car A and \[m\] is the magnification factor.

Complete step by step solution:
We know that the object distance (u) is always negative for a concave or convex mirror since an object is always positioned to the left side of the mirror as well as distances toward the left side of the mirror are always negative.
So, \[u = - 1.9{\text{ m}}\]
Focal length (f) = 10 cm
Here, we will convert the object distance into cm.
We know that \[1{\text{ m}} = 100{\text{ cm}}\]
So, \[u = - 1.9 \times 100 = - 190cm\]

By using magnification formula, we get
\[m = \dfrac{f}{{f - u}}\]
Thus, we get
\[m = \dfrac{{10}}{{10 - \left( { - 190} \right)}}\]
By simplifying, we get
\[m = \dfrac{{10}}{{10 + 190}}\]
\[\Rightarrow m = \dfrac{{10}}{{200}}\]
\[\Rightarrow m = \dfrac{1}{{20}}\]
Let us find the speed of car B when its driver observes car A in the side mirror.
\[{v_B} = - {m^2} \times \left( {{v_A}} \right)\]
\[\Rightarrow {v_B} = - {\left( {\dfrac{1}{{20}}} \right)^2} \times 45\]
\[\Rightarrow {v_B} = - \left( {\dfrac{{45}}{{400}}} \right)\]
\[\therefore {v_B} = - \left( {\dfrac{9}{{80}}} \right)\]
Hence, the speed of car B when its driver observes the car A in the side mirrors is \[ - \left( {\dfrac{9}{{80}}} \right){\text{ m/s}}\].

Therefore, the correct option is (B).

Note: Many students make mistakes in taking the proper sign of object distance while calculating the magnification factor. We know that the object distance is the distance between the point of incidence and also the object positioned in front of a mirror. Also, it is always negative according to the sign conventions of a mirror.