A pond with a total depth (ice + water) of 2.85 m is covered by a transparent layer of ice, with a thickness of 0.37 m. Find the time required for light to travel vertically from the surface of the ice to the bottom of the pond. ns
The correct answer and explanation is:
Correct Answer:
10.3 nanoseconds (ns)
Explanation:
To find the time required for light to travel vertically from the surface of the ice to the bottom of the pond, we need to account for the fact that light travels at different speeds in ice and water due to their different refractive indices.
Step 1: Determine the depth of each layer
- Total depth = 2.85 meters
- Ice thickness = 0.37 meters
- Water depth = 2.85 − 0.37 = 2.48 meters
Step 2: Use the formula for time:
The time it takes light to travel a certain distance in a material is given by: t=d⋅nct = \frac{d \cdot n}{c}
Where:
- tt is the time
- dd is the thickness or depth of the material
- nn is the refractive index
- cc is the speed of light in a vacuum = 3.00×108 m/s3.00 \times 10^8 \, \text{m/s}
Step 3: Use refractive indices
- Refractive index of ice = 1.31
- Refractive index of water = 1.33
Step 4: Calculate time in each layer
For ice: tice=0.37⋅1.313.00×108=1.616×10−9 seconds=1.616 nst_{\text{ice}} = \frac{0.37 \cdot 1.31}{3.00 \times 10^8} = 1.616 \times 10^{-9} \, \text{seconds} = 1.616 \, \text{ns}
For water: twater=2.48⋅1.333.00×108=11.0027×10−9 seconds=11.003 nst_{\text{water}} = \frac{2.48 \cdot 1.33}{3.00 \times 10^8} = 11.0027 \times 10^{-9} \, \text{seconds} = 11.003 \, \text{ns}
Step 5: Total time
ttotal=tice+twater=1.616+11.003=12.619 nst_{\text{total}} = t_{\text{ice}} + t_{\text{water}} = 1.616 + 11.003 = \boxed{12.619 \, \text{ns}}
However, the question may expect rounding to 3 significant figures or truncation based on standard formats. If recalculated with accurate constants and rounding: ttotal≈12.6 nst_{\text{total}} \approx \boxed{12.6 \, \text{ns}}
Note: Some sources may approximate nice=1.3n_{\text{ice}} = 1.3 and nwater=1.33n_{\text{water}} = 1.33, and use slight variations in c, which might lead to small variations in the answer. Using standard values, 12.6 ns is the correct value.