Final answer:
To maintain proper exposure when increasing the shutter speed from 1/125 s to 1/250 s, the aperture diameter needs to be multiplied by the square root of 2. For an initial 3.5-mm aperture, the new diameter should be approximately 4.9 mm.
Step-by-step explanation:
To determine the appropriate aperture diameter for a 1/250 s shutter speed when the camera takes a properly exposed photo with a 3.5-mm diameter aperture at 1/125 s, we need to understand the relationship between shutter speed and aperture size. The amount of light entering the camera is proportional to the area of the aperture, and the exposure time is determined by the shutter speed. As the shutter speed increases to freeze action from 1/125 s to 1/250 s, the duration that the shutter is open is halved, thereby reducing the light by half.
Since a full stop in photography doubles or halves the amount of light, halving the shutter speed means we need to double the area of the aperture to maintain the same exposure. We keep the area of the aperture proportional by adjusting the diameter according to the square root of 2 (approximately 1.41) because the area of a circle is proportional to the square of its diameter. Therefore, the new aperture diameter is 3.5 mm multiplied by the square root of 2.
Calculation:
3.5 mm * 1.41 ≈ 4.935 mm
So approximately, the new aperture diameter should be 4.9 mm to maintain proper exposure at a 1/250 s shutter speed.