The acceleration of a particle is directly proportional to the square of the time t. When t =0, the particle is at x =24 m. Knowing that at

Question

The acceleration of a particle is directly proportional to the square of the time t. When t =0, the particle is at x =24 m. Knowing that at t =6 s, x =96 m and v =18 m/s, express x and v in terms of t.

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Thu Thảo 3 years 2021-08-31T16:51:24+00:00 1 Answers 147 views 0

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  1. thuthuy
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    2021-08-31T16:52:56+00:00
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    Answer:

    x(t) = ⅟₁₀₈t⁴ + 10t + 24

    v(t) = ⅟₂₇t³ + 10

    Explanation:

    a(t) = C₁t²

    velocity is the integral of acceleration

    v(t) = ⅓C₁t³ + C₂

    position is the integral of velocity

    x(t) = (⅟₁₂C₁)t⁴ + C₂t + C₃

    x(0) = 24 = (⅟₁₂C₁)0⁴ + C₂0 + C₃

    C₃ = 24

    x(6) = 96 = (⅟₁₂C₁)6⁴ + C₂6 + 24

             72 = 108C₁ + 6C₂

             C₂ = 12 – 18C₁

    v(6) = 18 = ⅓C₁6³ + C₂

              18 = 72C₁ + C₂

              18 = 72C₁ + (12 – 18C₁)

               6 = 54C₁

               C₁ = 1/9

               C₂ = 12 – 18(1/9)

               C₂ = 10

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