A concave mirror forms an image of 20 cm high object on a screen placed 5.0 m away from the mirror. The height of the image is 50 cm. Find the focal length of the mirror and the distance between the mirror and the object.

Explanation

To solve the problem, we need to find the focal length of the concave mirror and the distance between the mirror and the object. We can use the mirror formula and the magnification formula for this purpose.

### Step 1: Identify the known values
- Height of the object (h_o) = 20 cm
- Height of the image (h_i) = -50 cm (negative because the image is inverted)
- Distance from the mirror to the screen (v) = 5.0 m = 500 cm (since we need to keep the units consistent)

### Step 2: Use the magnification formula
The magnification (m) is given by the formula:
\[ m = \frac{h_i}{h_o} = -\frac{v}{u} \]

Substituting the known values:
\[ m = \frac{-50}{20} = -2.5 \]

### Step 3: Relate magnification to object distance (u) and image distance (v)
From the magnification formula, we can write:
\[ -2.5 = -\frac{500}{u} \]

### Step 4: Solve for u
Rearranging the equation gives:
\[ 2.5 = \frac{500}{u} \]
\[ u = \frac{500}{2.5} = 200 \text{ cm} \]

### Step 5: Use the mirror formula to find the focal length (f)
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]

Substituting the values of u and v:
\[ \frac{1}{f} = \frac{1}{200} + \frac{1}{500} \]

### Step 6: Calculate the right-hand side
Finding a common denominator (which is 1000):
\[ \frac{1}{f} = \frac{5}{1000} + \frac{2}{1000} = \frac{7}{1000} \]

### Step 7: Solve for f
Taking the reciprocal gives:
\[ f = \frac{1000}{7} \approx 142.86 \text{ cm} \]

### Summary of Results
- The distance between the mirror and the object (u) = 200 cm
- The focal length of the mirror (f) ≈ 142.86 cm