A new unit (\(\alpha\)) of length is chosen such that it is equal to the speed of light in vacuum. What is the distance between Venus and Earth in terms of \(\alpha\) units if light takes 6 min. 40 s to cover this distance?
Step 1: Understanding the Concept:
The problem defines a custom unit of length $\alpha$ corresponding to the distance light travels in a specific, although implied, time unit (usually one second based on context). We need to calculate the distance travelled by light in a given amount of time using this new unit. Step 2: Key Formula or Approach:
Distance $d = v \times t$.
Given that the new unit $\alpha$ is defined as the distance light travels in 1 second, the speed of light is $v = 1 \alpha/\text{s}$.
Convert the given time entirely into seconds and apply the formula. Step 3: Detailed Explanation:
Let the new unit of length be $\alpha$.
The speed of light in vacuum is defined as $c = 1 \alpha \text{ per second}$.
The time taken for light to cover the distance between Venus and Earth is given as:
$t = 6 \text{ minutes } 40 \text{ seconds}$.
Convert the time into standard SI seconds:
$t = (6 \times 60) \text{ s} + 40 \text{ s} = 360 \text{ s} + 40 \text{ s} = 400 \text{ s}$.
Now, calculate the distance using the fundamental relation:
Distance = Speed $\times$ Time
$d = c \times t = (1 \alpha/\text{s}) \times 400 \text{ s} = 400 \alpha$. Step 4: Final Answer:
The distance is $400 \alpha$.
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