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 step by step, we will follow these instructions:

### Step 1: Identify the Given Information
- Height of the object (h_o) = 20 cm
- Height of the image (h_i) = -50 cm (negative because the image is inverted)
- Image distance (v) = -500 cm (since the image is real and on the same side as the object in a concave mirror)

### 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:
\[ -\frac{50}{20} = -\frac{-500}{u} \]

### Step 3: Calculate the Object Distance (u)
From the magnification equation:
\[ -\frac{5}{2} = \frac{500}{u} \]

Cross-multiplying gives:
\[ -5u = 1000 \]
\[ u = -200 \text{ cm} \]

### Step 4: Convert Object Distance to Meters
Since we need the distance in meters:
\[ u = -200 \text{ cm} = -2 \text{ m} \]

### Step 5: Use the Mirror Formula to Find the Focal Length (f)
The mirror formula is:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]

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

### Step 6: Calculate the Focal Length (f)
Finding a common denominator (1000):
\[ \frac{1}{f} = \frac{-5}{1000} + \frac{-2}{1000} \]
\[ \frac{1}{f} = \frac{-7}{1000} \]

Taking the reciprocal:
\[ f = -\frac{1000}{7} \approx -142.86 \text{ cm} \]
Converting to meters:
\[ f \approx -1.43 \text{ m} \]

### Summary of Results
- The object distance (u) is -2 m.
- The focal length (f) is approximately -1.43 m.