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Thread: Mechanics Speed Question

  1. #1
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    Exclamation Mechanics Speed Question

    A balloon moves vertically upwards in a straight line so that at time t seconds its acceleration, a metre per second is
    a=3t/200

    Initally the balloon has a speeed of 0.5m/s
    i)find the speed of the balloon when t=30sec
    ii)Find how far the balloon travels in the first 30sec.

    Any help much appreciated.
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  2. #2
    Member Danshader's Avatar
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    hi there,

    we have the following data

    $\displaystyle
    Acceleration,a = \frac{3t}{200}
    $
    $\displaystyle
    InitialSpeed, u= 0.5
    $
    $\displaystyle
    TimeTaken, t = 30
    $

    to find the final velocity we will use the formula
    $\displaystyle
    v = u+at
    $

    substituting values we have the final velocity can be obtained


    to find the distance traveled for this first 30 seconds use this formula with the given information:
    $\displaystyle
    s = ut+\frac{1}{2}at^2
    $

    $\displaystyle
    Acceleration,a = \frac{3t}{200}
    $
    $\displaystyle
    InitialSpeed, u= 0.5
    $
    $\displaystyle
    TimeTaken, t = 30
    $

    again by substituting the given value you should be able to get the required distance traveled ^-^ good luck.
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  3. #3
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    yes i have tried using those equationns previously but got answers of 13 etc. the answer i know is infact 7.25.help plz!!
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  4. #4
    Member Danshader's Avatar
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    ou ok i understand why it is not this method, the previous method was for constant acceleration, however in this case it is for varying acceleration:

    the right method to use is to consider:
    $\displaystyle
    \frac{d}{dt}v = a
    $

    $\displaystyle
    \frac{d}{dt}v = \frac{3t}{200}
    $

    $\displaystyle
    \int \frac{d}{dt}v = \int \frac{3t}{200}dt
    $

    $\displaystyle
    v = \frac{3t^2}{400} + c \longrightarrow(1)
    $

    when t = 0, u = 0.5, from (1);
    $\displaystyle
    u = \frac{3(0)^2}{400} + c
    $
    $\displaystyle
    c = 0.5
    $

    so the equation to determine the speed of the balloon after 30 seconds is:
    $\displaystyle
    v = \frac{3t^2}{400} + 0.5
    $

    $\displaystyle
    v = \frac{3(30)^2}{400} + 0.5
    $

    this will give you the answer of 7.25 ms^(-1)

    for the second part we can also observe just like in the case of finding the velocity:

    $\displaystyle
    \frac{d}{dt}S = v
    $

    $\displaystyle
    \int \frac{d}{dt}S =\int vdt
    $

    $\displaystyle
    \int \frac{d}{dt}S =\int \frac{3t^2}{400} + 0.5dt
    $

    $\displaystyle
    S = \frac{3t^3}{1200} + 0.5t + k \longrightarrow(2)
    $

    at t = 0, S = 0, therefore from equation (2):
    $\displaystyle
    S = \frac{3t^3}{1200} + 0.5t + k
    $

    $\displaystyle
    0 = \frac{3(0)^3}{1200} + 0.5(0) + k
    $
    $\displaystyle
    k = 0
    $

    therefore the general equation of the displacement, S is:

    $\displaystyle
    S = \frac{3t^3}{1200} + 0.5t
    $
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