Introduction:
In this scenario, a man is taking an escalator to reach his destination. The escalator takes him to his destination in 1 minute, while he can cover the same distance in 3 minutes if he walks with a uniform speed. The question asks how many minutes it takes for the escalator to reach its acceleration.

Explanation:
To understand the time it takes for the escalator to reach its acceleration, we need to analyze the motion of the man on the escalator.

Uniform Speed:
When the man walks with a uniform speed, he can cover the distance in 3 minutes. This means that his speed remains constant throughout the journey. Let's assume the distance is d and the man's speed is v. Therefore, we can write the equation:

Distance = Speed × Time
d = v × 3

Escalator Speed:
The escalator takes the man to his destination in 1 minute. This means that the speed of the escalator combined with the man's walking speed results in a faster overall speed. Let's assume the speed of the escalator is e. Therefore, the equation can be written as:

Distance = (Speed + Escalator Speed) × Time
d = (v + e) × 1

Acceleration:
Acceleration is the rate of change of velocity. Since the man's speed remains constant while walking, the acceleration is associated with the speed of the escalator. To find the time it takes for the escalator to reach its acceleration, we need to determine the change in speed.

We know that:
Speed = Distance / Time

By rearranging the equation, we can find the change in speed:
Change in Speed = Speed with Escalator - Speed without Escalator
Change in Speed = (v + e) - v
Change in Speed = e

So, the acceleration of the escalator is equal to the speed of the escalator (e).

Conclusion:
In this scenario, the time it takes for the escalator to reach its acceleration is 0 minutes. This is because the escalator starts at its maximum speed and does not require any additional time to reach its acceleration.

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