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Math Help - Can the average velocity and delta velocity ever be the same?

  1. #1
    CrazziGirl23
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    Question Can the average velocity and delta velocity ever be the same?

    I think they can be but I'm not sure.. but I also need to know why. I was thinking if acceleration was the same then delta velocity and average velocity could be the same? Help please!
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by CrazziGirl23 View Post
    I think they can be but I'm not sure.. but I also need to know why. I was thinking if acceleration was the same then delta velocity and average velocity could be the same? Help please!
    The "acceleration was the same" as what?

    Of course \Delta v and v can be the same. Given the appropriate acceleration and time interval this could certainly happen. Of course, it isn't likely to happen in an arbitrary situation.

    -Dan
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    Quote Originally Posted by CrazziGirl23 View Post
    I think they can be but I'm not sure.. but I also need to know why. I was thinking if acceleration was the same then delta velocity and average velocity could be the same? Help please!
    Even if the acceleration does not remain the same or constant, the delta velocity can never be the same as the average velocity.

    That is if by "delta velocity" you mean the difference between two velocities.
    Say you have Vo and V1.
    delta V = V1 -Vo
    average V = (Vo +V1)/2
    They can never be the same.
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    Quote Originally Posted by ticbol View Post
    Even if the acceleration does not remain the same or constant, the delta velocity can never be the same as the average velocity.

    That is if by "delta velocity" you mean the difference between two velocities.
    Say you have Vo and V1.
    delta V = V1 -Vo
    average V = (Vo +V1)/2
    They can never be the same.
    That's not true. You are talking about arithmetic mean of velocities which is different than average velocity...

    The average velocity in the interval [t1, t2] is defined as:
    \frac{1}{t_2 - t_1} \int_{t_1}^{t_2} v(t) dt

    I can easily think of an example where delta v and <v> are the same...

    Let's consider motion in interval [0, 2s] on the straight line, with constant accelelation 1 \frac{m}{s^2}. Let's assume that initial velocity is equal 1 \frac{m}{s}. One can easily check that: \Delta v = <v> = 2 \frac{m}{s}
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    Quote Originally Posted by albi View Post
    That's not true. You are talking about arithmetic mean of velocities which is different than average velocity...

    The average velocity in the interval [t1, t2] is defined as:
    \frac{1}{t_2 - t_1} \int_{t_1}^{t_2} v(t) dt

    I can easily think of an example where delta v and <v> are the same...

    Let's consider motion in interval [0, 2s] on the straight line, with constant accelelation 1 \frac{m}{s^2}. Let's assume that initial velocity is equal 1 \frac{m}{s}. One can easily check that: \Delta v = <v> = 2 \frac{m}{s}
    I see.

    So I was wrong. Okay.

    Only in your example.

    In the average velocity as shown in your integration, [which is correct, by the way], I don't know.
    Can you also show me an example there?
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by ticbol View Post
    Even if the acceleration does not remain the same or constant, the delta velocity can never be the same as the average velocity.

    That is if by "delta velocity" you mean the difference between two velocities.
    Say you have Vo and V1.
    delta V = V1 -Vo
    average V = (Vo +V1)/2
    They can never be the same.
    Let's use your example instead, since I suspect you will find this more persuasive.

    \Delta v = v_1 - v_0
    \bar{v} = \frac{v_0 + v_1}{2}

    Set \Delta v = \bar{v}. We may solve the resulting equation for v_1 in terms of v_0:

    v_1 - v_0 = \frac{v_0 + v_1}{2}

    Thus
    v_1 = 3v_0

    So we have the condition for a general situation such that \Delta v and \bar{v} are the same.

    -Dan
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    Quote Originally Posted by topsquark View Post
    Let's use your example instead, since I suspect you will find this more persuasive.

    \Delta v = v_1 - v_0
    \bar{v} = \frac{v_0 + v_1}{2}

    Set \Delta v = \bar{v}. We may solve the resulting equation for v_1 in terms of v_0:

    v_1 - v_0 = \frac{v_0 + v_1}{2}

    Thus
    v_1 = 3v_0

    So we have the condition for a general situation such that \Delta v and \bar{v} are the same.

    -Dan
    Yes. That's true. Nevertheless we usually define average velocity (or speed?) using the integral i wrote previously... If we observe that the distanse made by moving body is equal to:

    s = \int_{t_1}^{t_2} v(t) dt

    We can get a simple relation between distanse and average velocity:

    s = \frac{<v>}{\Delta t}

    If you like arithmetic mean you should observe that "average velocity" is equal to arithmetic mean if accelelation is constant.
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  8. #8
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by albi View Post
    Nevertheless we usually define average velocity (or speed?) using the integral i wrote previously...
    Actually that isn't true. The integral is indeed the Mathematical average of the velocity function, but the general definition of an average velocity in Physics is
    \bar{v} = \frac{x_f - x_i}{t_f - t_i}
    where the x's are displacements.

    I also agree that Ticbol's average velocity is not at all general, as it presupposes a constant acceleration in order to have that form.

    -Dan
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    Frankly speaking it is the matter of terminology....

    Unfortunatelly I'm not very familiar with physics, english terminology...it can cause some problems...

    In polish terminology there is the distinction between "velocity" (prędkość) and "speed" (szybkość)...

    As "speed" I consider modulus of "velocity" - scalar quantity. "Velocity" is of a course vector...

    The average "speed" is defined with the above integral...

    What's with the average "velocity", it is analogous, so it is defined with the integral:

     \frac{1}{t_2-t_1} \int_{t_1}^{t_2} \vec{v} dt

    Obviously this definition is equivalent to yours because:

     \int_{t_1}^{t_2} \vec{v} dt = \int_{t_1}^{t_2} d \vec{r} = \vec{r_1} - \vec{r_0}
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  10. #10
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by albi View Post
    Frankly speaking it is the matter of terminology....

    Unfortunatelly I'm not very familiar with physics, english terminology...it can cause some problems...

    In polish terminology there is the distinction between "velocity" (prędkość) and "speed" (szybkość)...

    As "speed" I consider modulus of "velocity" - scalar quantity. "Velocity" is of a course vector...

    The average "speed" is defined with the above integral...

    What's with the average "velocity", it is analogous, so it is defined with the integral:

     \frac{1}{t_2-t_1} \int_{t_1}^{t_2} \vec{v} dt

    Obviously this definition is equivalent to yours because:

     \int_{t_1}^{t_2} \vec{v} dt = \int_{t_1}^{t_2} d \vec{r} = \vec{r_1} - \vec{r_0}
    Heh. My bad. I guess I had a brain fart for a moment there...

    -Dan
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