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16 November, 07:36

The acceleration of a Maserati is proportional to the difference between 250 km/h and the velocity of this sports car. a) Write the differential equation governing the velocity. Use k for the proportionalty constant (assume k>0). b) Solve the differential equation for v (t) such that v (0) = 0

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  1. 16 November, 09:00
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    the equation for v is v = 250 km/h * [1 - e^ (-kt) ]

    Step-by-step explanation:

    since the acceleration "a" is the rate of change of velocity "v" with respect to the time "t", we have

    a = dv/dt = k * (250 - v)

    thus

    dv / (250 - v) = k*dt

    if we proceed then to the indefinite integration in both sides

    ∫dv / (250 - v) = ∫k*dt

    - ln (250-v) = k*t + C, where C = constant

    if we know that at t=0, v (t=0) = 0, then

    - ln (250-0) = k*0 + C

    - ln 250 = C

    replacing C in the original equation:

    - ln (250-v) = k*t - ln 250

    ln [ (250-v) / 250] = - k*t

    v = 250 km/h * [1 - e^ (-kt) ]
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