Where should a person stand straight from the pole of a convex mirror of focal length 2.0 m on its axis, so that the image formed become half of his original height?

Options

A. `-2.60` m
B. `-4.0` m
C. `-0.5` m
D. `-2.0` m

Answer

To solve the problem, we need to determine the position of a person standing in front of a convex mirror such that the image formed is half the height of the person. Here’s the step-by-step solution:

### Step 1: Understand the mirror formula and magnification
The mirror formula for a convex mirror is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
where:
- \( f \) is the focal length of the mirror (positive for a convex mirror),
- \( v \) is the image distance (positive for virtual images formed by convex mirrors),
- \( u \) is the object distance (negative as per sign convention).

The magnification \( m \) is given by:
\[ m = \frac{h'}{h} = -\frac{v}{u} \]
where:
- \( h' \) is the height of the image,
- \( h \) is the height of the object.

### Step 2: Set up the magnification condition
We want the image to be half the height of the object, which means:
\[ m = \frac{1}{2} \]
Thus, we can write:
\[ -\frac{v}{u} = \frac{1}{2} \]
From this, we can express \( v \) in terms of \( u \):
\[ v = -\frac{1}{2} u \]

### Step 3: Substitute \( v \) into the mirror formula
Now, we substitute \( v \) in the mirror formula:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting \( v = -\frac{1}{2} u \) and \( f = 2 \, m \):
\[ \frac{1}{2} = \frac{1}{-\frac{1}{2}u} + \frac{1}{u} \]

### Step 4: Simplify the equation
This can be rewritten as:
\[ \frac{1}{2} = -\frac{2}{u} + \frac{1}{u} \]
Combining the terms on the right gives:
\[ \frac{1}{2} = -\frac{2 - 1}{u} = -\frac{1}{u} \]
Thus, we have:
\[ \frac{1}{u} = -\frac{1}{2} \]
Taking the reciprocal gives:
\[ u = -2 \, m \]

### Step 5: Conclusion
The negative sign indicates that the object (the person) is positioned 2 meters in front of the mirror. Therefore, the person should stand **2 meters from the pole of the convex mirror**.

### Final Answer
The person should stand 2 meters from the pole of the convex mirror.

---