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

Explanation

To solve the problem of where a person should stand in front of a convex mirror with a focal length of 2.0 m so that the image formed is half of his original height, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- Focal length of the convex mirror, \( f = +2.0 \, \text{m} \) (positive for convex mirrors).
- Magnification, \( m = \frac{h'}{h} = \frac{1}{2} \) (the image height is half of the object height).

2. **Use the Magnification Formula**:
The magnification \( m \) for mirrors is given by the formula:
\[ m = -\frac{v}{u} \]
where \( v \) is the image distance and \( u \) is the object distance. For a convex mirror, the image is virtual, erect, and diminished, so \( m \) will be positive.

3. **Relate Magnification to Object Distance**:
Since we know that \( m = \frac{1}{2} \), we can set up the equation:
\[ \frac{1}{2} = -\frac{v}{u} \]
Rearranging gives:
\[ v = -\frac{u}{2} \]

4. **Use the Mirror Formula**:
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting \( f = 2.0 \, \text{m} \) and \( v = -\frac{u}{2} \):
\[ \frac{1}{2} = \frac{1}{-\frac{u}{2}} + \frac{1}{u} \]

5. **Simplify the Equation**:
Replacing \( \frac{1}{-\frac{u}{2}} \) with \(-\frac{2}{u}\):
\[ \frac{1}{2} = -\frac{2}{u} + \frac{1}{u} \]
Combine the terms on the right:
\[ \frac{1}{2} = -\frac{2}{u} + \frac{1}{u} = -\frac{2 - 1}{u} = -\frac{1}{u} \]

6. **Solve for \( u \)**:
Rearranging gives:
\[ u = -2 \, \text{m} \]

7. **Interpret the Result**:
The negative sign indicates that the object (the person) must be placed 2 m in front of the mirror, which is consistent with the convention that object distances are negative for real objects in mirror formulas.

### Final Answer:
The person should stand 2.0 m in front of the convex mirror.