To model the fly's movement with respect to time, we can use a piecewise function. Let's denote the distance of the fly from the lightbulb at time \( t \) (in seconds) as \( d(t) \).
Piecewise function: Assuming we break the movement of the fly into distinct segments, we can describe its flight as follows:
- From \( t = 0 \) to \( t = 1 \) second, the fly moves towards the lightbulb.
- From \( t = 1 \) to \( t = 2 \) seconds, the fly moves away from the lightbulb.
- From \( t = 2 \) to \( t = 3 \) seconds, the fly again moves towards the lightbulb and hits it.
- From \( t = 3 \) to \( t = 4 \) seconds, the fly finally flies away from the lightbulb.
Defining the segments of \( d(t) \):
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For \( 0 \leq t < 1 \): Assume the fly moves towards the bulb linearly from 2 meters to 0 meters: \[ d(t) = 2 - 2t \quad (0 \leq t < 1) \] At \( t = 0, d(0) = 2 \) and at \( t = 1, d(1) = 0\).
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For \( 1 \leq t < 2 \): The fly moves away from the bulb linearly from 0 meters to 2 meters: \[ d(t) = 2t - 2 \quad (1 \leq t < 2) \] At \( t = 1, d(1) = 0 \) and at \( t = 2, d(2) = 2\).
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For \( 2 \leq t < 3 \): The fly moves towards the bulb again, hitting it at \( t = 3\): \[ d(t) = 2 - 2(t - 2) = 6 - 2t \quad (2 \leq t < 3) \] At \( t = 2, d(2) = 2 \) and at \( t = 3, d(3) = 0\).
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For \( 3 \leq t < 4 \): After hitting the bulb, it flies away, from 0 to 2 meters again: \[ d(t) = 2(t - 3) \quad (3 \leq t < 4) \] At \( t = 3, d(3) = 0\) and at \( t = 4, d(4) = 2\).
Complete Piecewise Function:
Combining all segments, we get: \[ d(t) = \begin{cases} 2 - 2t & \text{if } 0 \leq t < 1 \ 2t - 2 & \text{if } 1 \leq t < 2 \ 6 - 2t & \text{if } 2 \leq t < 3 \ 2(t - 3) & \text{if } 3 \leq t < 4 \ \end{cases} \]
Verification of Requirements:
- At \( t = 0 \): \( d(0) = 2 \) (2 meters from the bulb)
- At \( t = 1 \): \( d(1) = 0 \) (fly reaches the bulb)
- At \( t = 2 \): \( d(2) = 2 \) (fly 2 meters away again)
- At \( t = 3 \): \( d(3) = 0 \) (fly hits the bulb)
- At \( t = 4 \): \( d(4) = 2 \) (fly is 2 meters away again)
The piecewise function describes the fly's journey appropriately and satisfies all the situational requirements.