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Distance between two positions from the velocities and accelerations there

The same question, asked 3 times: 11 May 2023 S1, 16 May 2023 S2, 20 Apr 2025 S2. Every wording below, oldest first.

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  1. Question 1

    11 May 2023 S1

    46
    A particle performs linear S.H.M. At a particular instant, velocity of the particle is 'u' and acceleration is 'a1a_1' while at another instant velocity is 'V' and acceleration is 'a2a_2' where 0<a1<a20 < a_1 < a_2. The distance between the two positions is

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  2. Question 2

    16 May 2023 S2

    43
    A particle performs linear SHM. At a particular instant, velocity of the particle is uu and acceleration is α\alpha while at another instant, velocity is vv and acceleration is β\beta (0<α<β0 < \alpha < \beta). The distance between the two positions is

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  3. Question 3

    20 Apr 2025 S2

    33
    A particle executes a linear S.H.M. In two of its positions the velocities are V1,V2V_{1},V_{2} and accelerations are a1a_{1} and a2a_{2} respectively (0<a1<a2)\left( 0 <a_{1}<a_{2} \right). The distance between the positions is

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