A source of light is at 10 cm from a convex mirror and is then moved to a distance of 2 cm from the mirror. How much does the image move if the radius of curvature of the mirror is 4.8 cm?

Options

A. 0.95 cm
B. 0.85 cm
C. 0.75 cm
D. 0.65 cm

Explanation

To solve the problem step by step, we will use the mirror formula and the information provided in the question.

### Step 1: Identify the Given Data
- Radius of curvature (R) = 4.8 cm
- Initial object distance (U1) = -10 cm (negative because it is in front of the mirror)
- Final object distance (U2) = -2 cm (also negative for the same reason)

### Step 2: Calculate the Focal Length (F)
The focal length (F) of a convex mirror is given by:
\[ F = \frac{R}{2} \]
Substituting the value of R:
\[ F = \frac{4.8 \, \text{cm}}{2} = 2.4 \, \text{cm} \]
Since it is a convex mirror, the focal length is positive:
\[ F = +2.4 \, \text{cm} \]

### Step 3: Use the Mirror Formula to Find Image Distances
The mirror formula is:
\[ \frac{1}{F} = \frac{1}{V} + \frac{1}{U} \]
Rearranging gives:
\[ \frac{1}{V} = \frac{1}{F} - \frac{1}{U} \]

#### Step 3.1: Calculate V1 for U1 = -10 cm
Substituting the values for U1:
\[ \frac{1}{V1} = \frac{1}{2.4} - \frac{1}{-10} \]
Finding a common denominator (which is 24):
\[ \frac{1}{V1} = \frac{10}{24} + \frac{2.4}{24} = \frac{10 + 2.4}{24} = \frac{12.4}{24} \]
Thus,
\[ V1 = \frac{24}{12.4} \approx 1.935 \, \text{cm} \]

#### Step 3.2: Calculate V2 for U2 = -2 cm
Now substituting the values for U2:
\[ \frac{1}{V2} = \frac{1}{2.4} - \frac{1}{-2} \]
Finding a common denominator (which is 24):
\[ \frac{1}{V2} = \frac{10}{24} + \frac{12}{24} = \frac{22}{24} \]
Thus,
\[ V2 = \frac{24}{22} \approx 1.09 \, \text{cm} \]

### Step 4: Calculate the Movement of the Image
To find how much the image has moved, we calculate the difference between V1 and V2:
\[ \text{Movement} = |V1 - V2| = |1.935 - 1.09| \approx 0.845 \, \text{cm} \]

### Final Answer
The image moves approximately 0.845 cm.

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