To solve this problem, we use the mirror equation for a convex mirror, which is given by:
[tex]\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \][/tex]
where:
- [tex]\( f \)[/tex] is the focal length of the mirror,
- [tex]\( v \)[/tex] is the image distance from the mirror,
- [tex]\( u \)[/tex] is the object distance from the mirror.
For a convex mirror:
1. The focal length ([tex]\( f \)[/tex]) is positive: [tex]\( f = 1 \)[/tex] m.
2. The object distance ([tex]\( u \)[/tex]) is negative as per the sign conventions used in mirror problems.
Let's denote the nearer end of the pole as [tex]\( A \)[/tex] and the farther end as [tex]\( B \)[/tex].
Step 1: Determining the image distance for end [tex]\( A \)[/tex]
The object distance for end [tex]\( A \)[/tex] is:
[tex]\[ u_A = -2 \, \text{m} \][/tex]
Using the mirror equation:
[tex]\[ \frac{1}{f} = \frac{1}{v_A} + \frac{1}{u_A} \][/tex]
Substituting the known values:
[tex]\[ \frac{1}{1} = \frac{1}{v_A} + \frac{1}{-2} \][/tex]
Solving for [tex]\( v_A \)[/tex]:
[tex]\[ 1 = \frac{1}{v_A} - 0.5 \][/tex]
[tex]\[ \frac{1}{v_A} = 1.5 \][/tex]
[tex]\[ v_A = \frac{1}{1.5} \][/tex]
[tex]\[ v_A = \frac{2}{3} \][/tex]
[tex]\[ v_A = 0.6667 \, \text{m} \][/tex]
Step 2: Determining the image distance for end [tex]\( B \)[/tex]
The object distance for end [tex]\( B \)[/tex] is:
[tex]\[ u_B = -(2 + 4) = -6 \, \text{m} \][/tex]
Using the mirror equation:
[tex]\[ \frac{1}{f} = \frac{1}{v_B} + \frac{1}{u_B} \][/tex]
Substituting the known values:
[tex]\[ \frac{1}{1} = \frac{1}{v_B} + \frac{1}{-6} \][/tex]
Solving for [tex]\( v_B \)[/tex]:
[tex]\[ 1 = \frac{1}{v_B} - \frac{1}{6} \][/tex]
[tex]\[ \frac{1}{v_B} = 1 + \frac{1}{6} \][/tex]
[tex]\[ \frac{1}{v_B} = \frac{7}{6} \][/tex]
[tex]\[ v_B = \frac{6}{7} \][/tex]
[tex]\[ v_B = 0.8571 \, \text{m} \][/tex]
Step 3: Calculating the length of the image
The length of the image is the absolute difference between image distances [tex]\( v_A \)[/tex] and [tex]\( v_B \)[/tex]:
[tex]\[ \text{Image length} = |v_B - v_A| \][/tex]
Substituting the image distances:
[tex]\[ \text{Image length} = |0.8571 - 0.6667| \][/tex]
[tex]\[ \text{Image length} = 0.1905 \, \text{m} \][/tex]
Therefore, the length of the image of the pole is approximately [tex]\( 0.1905 \)[/tex] meters.
