At what distance (in m) from a convex mirror of focal length 2.5 m should a boy stand so that his image has a height equal to half of his height? The principal axis of the mirror is perpendicular to the height of the boy.

Explanation

To solve the problem, we need to find the distance at which a boy should stand from a convex mirror so that his image height is half of his actual height. Here are the steps to arrive at the solution:

### Step 1: Identify Given Data
- Focal length of the convex mirror (F) = +2.5 m (positive because it's a convex mirror)
- Height of the image (h_i) = 1/2 * height of the object (h_o)

### Step 2: Use the Magnification Formula
The magnification (m) is given by the ratio of the height of the image to the height of the object:
\[ m = \frac{h_i}{h_o} \]
Given that the image height is half of the object height, we can write:
\[ m = \frac{1}{2} \]

### Step 3: Relate Magnification to Object and Image Distances
The magnification can also be expressed in terms of the object distance (u) and image distance (v):
\[ m = -\frac{v}{u} \]
From the previous step, we have:
\[ \frac{1}{2} = -\frac{v}{u} \]
This implies:
\[ v = -\frac{u}{2} \]

### Step 4: Use the Mirror Formula
The mirror formula relates the object distance (u), image distance (v), and focal length (F):
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting \( v = -\frac{u}{2} \) into the mirror formula:
\[ \frac{1}{2.5} = \frac{1}{-\frac{u}{2}} + \frac{1}{u} \]

### Step 5: Simplify the Equation
Now, we can simplify the equation:
\[ \frac{1}{2.5} = -\frac{2}{u} + \frac{1}{u} \]
Combining the terms on the right side:
\[ \frac{1}{2.5} = -\frac{2 - 1}{u} \]
\[ \frac{1}{2.5} = -\frac{1}{u} \]

### Step 6: Solve for Object Distance (u)
Now, we can solve for u:
\[ u = -2.5 \]
Since distance cannot be negative, we take the absolute value:
\[ u = 2.5 \, \text{m} \]

### Final Answer
The boy should stand at a distance of **2.5 meters** from the convex mirror.