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physics

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a car of mass 'm'starts from rest and is driven with constant power.after some time it covers a distance 'x'.what should be the speed of car?how we can find the speed of car?

  • physics -

    power = force * speed

    p = F v
    so
    F = p/v

    F = m a = p v
    so
    a = (p/m) v
    or
    dv/dt = (p/m) v

    try form
    v = b e^kt + c
    dv/dt = b k e^kt
    so k = p/m
    so

    v = b e^(p/m)t + c
    at t = 0, v = 0
    0 = b + c
    so
    v = b e^(p/m) t - b

    at time t, it is at x
    this requires solving differential equation of form
    dx/dt = b e^kt - b
    x = (b/k) e^kt - bt + c
    at t = 0, x = 0
    0 = (b/k) + c
    so c = -b/k = -bm/p
    x = (bm/k) e^(p/m)t - b t -bm/p
    given an x at time t, solve for b

  • error repair -

    power = force * speed

    p = F v
    so
    F = p/v

    F = m a = p/ v
    so
    a = p/(m v)
    or
    v dv/dt = (p/m)

    v dv = (p/m) dt
    v^2/2 = (p/m) t + constant which is zero
    v^2 = (2p/m)t
    v = (2 p t/m)^.5

  • physics -

    dx/dt = (2 p /m)^.5 t^.5
    let k = 2 p/m

    dx/dt = k t^.5

    dx = k t^.5 dt

    x = (k/1.5) t^1.5 + c
    when t = 0, x = 0 so c = 0

    x = (4 p/3m) t^1.5

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