At what distance from a convex mirror of focal length 2.5 m should as boy stand so that his image has a height equal to half the original height?

Options

A. 2.5 m from the mirror
B. 5 m from the mirror
C. 7.5 m from the mirror
D. 10 m from the mirror

Answer

To solve the problem, we need to find the distance at which a boy should stand in front of a convex mirror with a focal length of 2.5 m so that his image height is half of his original height.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- Focal length of the convex mirror (f) = +2.5 m (positive because focal length of convex mirrors is taken as positive).
- The image height (h') is half of the original height (h). Therefore, if we let the original height be h, then h' = h/2.

2. **Determine the Magnification (m):**
- Magnification (m) is given by the formula:
\[ m = \frac{h'}{h} = \frac{\frac{h}{2}}{h} = \frac{1}{2} \]
- For mirrors, magnification can also be expressed in terms of image distance (v) and object distance (u):
\[ m = -\frac{v}{u} \]
- Since the magnification is positive for virtual images formed by convex mirrors, we can write:
\[ \frac{1}{2} = -\frac{v}{u} \]

3. **Relate Image Distance (v) and Object Distance (u):**
- From the magnification equation, we can express v in terms of u:
\[ v = -\frac{1}{2}u \]

4. **Use the Mirror Formula:**
- The mirror formula for a convex mirror is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
- Substituting f = 2.5 m into the formula:
\[ \frac{1}{2.5} = \frac{1}{v} + \frac{1}{u} \]

5. **Substitute v in terms of u:**
- Substitute \( v = -\frac{1}{2}u \) into the mirror formula:
\[ \frac{1}{2.5} = \frac{1}{-\frac{1}{2}u} + \frac{1}{u} \]
- This simplifies to:
\[ \frac{1}{2.5} = -\frac{2}{u} + \frac{1}{u} \]
- Combine the fractions:
\[ \frac{1}{2.5} = -\frac{2 - 1}{u} = -\frac{1}{u} \]

6. **Solve for u:**
- Rearranging gives:
\[ u = -2.5 \text{ m} \]
- The negative sign indicates that the object (the boy) is placed in front of the mirror, which is the expected position.

7. **Conclusion:**
- The boy should stand 2.5 m in front of the convex mirror.

### Final Answer:
The boy should stand 2.5 m from the convex mirror.