When a man moves toward a plane mirror at a speed of 6m/s, the distance the man and his image increased at a speed of?

The question as written has a wording problem. It should read: when a man moves toward a plane mirror at 6 m/s, the distance between the man and his image decreases at what rate? The word "increased" appears to be a typo, but the rate is the same either way.

1. Analyze the question We need the rate at which the separation between the man and his image changes as he moves toward the mirror.

2. Concept In a plane mirror, the image is as far behind the mirror as the object is in front of it. If the man is a distance d from the mirror, his image is also d behind it, so the man-to-image distance is 2d.

3. Solution

• Man's distance from the mirror: d
• Image's distance behind the mirror: d
• Separation between man and image: S = 2d

If the man moves toward the mirror at 6 m/s, then d changes at 6 m/s. So:

S = 2d, so dS/dt = 2 × (dd/dt) = 2 × 6 = 12 m/s

Another way to see it: the man approaches the mirror at 6 m/s and the image approaches the mirror at 6 m/s from the other side. Their relative speed is 6 + 6 = 12 m/s.

4. Answer The man and his image change their separation at 12 m/s (closing in on each other, or increasing if the man moves away).

5. Tip For a plane mirror, the speed of the image relative to the object is twice the object's speed (when the object moves perpendicular to the mirror). Whenever you see "speed of the image relative to the man," double the man's speed. But if the question asks about the image's speed relative to the mirror, it is the same as the man's speed (6 m/s).