Step 1: Mirror Formula and Focal Length Calculation
The mirror formula is expressed as: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] where \(f\) is the focal length, \(u\) is the object distance, and \(v\) is the image distance. The focal length is half the radius of curvature, \(f = \frac{R}{2}\). Given a radius of curvature \(R = 40 \, \text{cm}\), the focal length is: \[ f = \frac{40}{2} = 20 \, \text{cm} \]
Step 2: Image Distance Determination
For object A at \(u_1 = 15 \, \text{cm}\), the image distance \(v_1\) is calculated using the mirror formula: \[ \frac{1}{20} = \frac{1}{v_1} + \frac{1}{15} \] Rearranging to solve for \(v_1\): \[ \frac{1}{v_1} = \frac{1}{20} - \frac{1}{15} = \frac{3 - 4}{60} = -\frac{1}{60} \quad \Rightarrow \quad v_1 = -60 \, \text{cm} \] A negative image distance signifies a virtual image formed 60 cm behind the mirror.
For object B at \(u_2 = 25 \, \text{cm}\), the image distance \(v_2\) is determined similarly: \[ \frac{1}{20} = \frac{1}{v_2} + \frac{1}{25} \] Solving for \(v_2\): \[ \frac{1}{v_2} = \frac{1}{20} - \frac{1}{25} = \frac{5 - 4}{100} = \frac{1}{100} \quad \Rightarrow \quad v_2 = 100 \, \text{cm} \] A positive image distance indicates a real image formed 100 cm in front of the mirror.
Step 3: Distance Between Images Calculation
The distance separating the two images is the absolute difference between their image distances: \[ \text{Distance between images} = |v_2 - v_1| = |100 \, \text{cm} - (-60 \, \text{cm})| = |100 + 60| = 160 \, \text{cm} \]
Final Answer: The distance between the formed images is \(160 \, \text{cm}\).