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Math Help - Predicting Total Movement after ( X ) Time

  1. #1
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    Predicting Total Movement after ( X ) Time

    Hi
    I have a moving Object.
    - There are two general forces affecting the Objects movement:
    --> The Objects own movement: ( Vi / Ln( D ) ) * ( D^T - 1 )
    --> Wind: ( T / 2 ) * ( T - 1 ) * W
    - The Object only moves in the horizontal direction, left or right, same with Wind.
    ( Time is Integer, all other values are potential floats )

    ...
    So i now have an expression describing the Objects Total Movement:
    ( Vi / Ln( D ) ) * ( D^T - 1 ) + ( T / 2 ) * ( T - 1 ) * W

    I want to know how far the Object moves before becomming lower or equal to Target Total Speed ( TTS )
    ( Vi / Ln( D ) ) * ( D^T - 1 ) + ( T / 2 ) * ( T - 1 ) * W < TTS

    All values except Time ( T ) are known.
    It is worth mentioning that there is generaly only a solution to this when Objects Initial Speed ( Vi ) and Wind Power ( W ) work in opposite directions ( +Vi -W or -Vi +W ).

    What am i to do next, do i need to solve for X/ find Time before i can continue with this problem?
    Am i on the right track here?
    What is the next step?
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: Predicting Total Movement after ( X ) Time

    Please clarify a few things. From previous posts I assume that D is a constant (not distance) between 0 and 1, and T = time, correct? You have the factor (T-1) in the expression for wind speed - does that mean that from T=0 to T=1 the wind blows in one direction, and after T=1 the wind blows in the other (i.e. the sign of W changes from negative to positive)? Also each of the two expressions you provided are velocities (not power), right? Finally, I trust you ar aware that the W component completely swamps out the V_i component after just a few seconds.
    Last edited by ebaines; May 14th 2013 at 06:51 AM.
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    Re: Predicting Total Movement after ( X ) Time

    Please clarify a few things. From previous posts I assume that D is a constant (not distance) between 0 and 1, and T = time, correct?

    This is the expression reflecting my Objects movement/ how far my Object moves:
    ( Vi / Ln( D ) ) * ( D^T - 1 ) + ( T / 2 ) * ( T - 1 ) * W

    ( only Time is integer, rest are potential floats )
    Vi = Objects InitialVelocity and direction ( Is this a Vector? ) ( + Object is moving right, - Object is moving left )
    D = Decelleration ( constant number, between 0 and 1 )
    T = Time ( integer only )
    W = Wind Power and Direction ( Is this a Vector? ) ( + Wind blows right, - Wind blows left )
    TTS = Target Total Velocity ( I need an expression returning how far Object has moved before reaching this Velocity )

    Also each of the two expressions you provided are velocities (not power), right?
    I believe they are Velocities, i dont realy know the difference between a Velocity and a Power.

    However Vi and W, as well as the return/ result of both parts of the whole expression, represents a velociti and a direction ( + right - left ), which makes me think they are Vectors, but im not sure ( im not too good at this stuff yet ).

    Finally, I trust you ar aware that the W component completely swamps out the V_i component after just a few seconds.
    Yep, i know the W component can be quite prominent, but it realy depends on the value of Vi and W.
    As you say, W might very well quickly outswamp Vi and its Decelleration.
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    MHF Contributor ebaines's Avatar
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    Re: Predicting Total Movement after ( X ) Time

    OK, I reviewed the earlier posts that generated these equations. They come from the following conditions:

    1. The first part of the equation comes from  v(t) = V_i D^t where D is a damping coefficient bewteen 0 and 1. Hence as time increase the velocity decreases. The distance covered in time t is  d(t) = \frac {V_i}{\ln D} (D^t -1)

    2. The second part comes from simple linear acceleration: v = wt , where w is the acceleration. The distance covered due to this acceleratio is d(t) = \frac 1 2 w t^2

    Hence the total velocity is:

    (1)  v(t) = V_iD^t + wt

    and the total distance is:

    (2) d(t) = \frac {V_i}{\ln D} (D^t -1) +\frac 1 2 w t^2.

    What you are asking is to calculate the time d when v= V_Target. This requires solving equation (1) to find t for when v = v_target, and then use that value of t in equation (2) to find the distance covered. Unfortunately there is no closed form solution to this. The best approach is to use a numerical technique to solve for t, then put that into the second equation.
    Last edited by ebaines; May 14th 2013 at 11:26 AM.
    Thanks from topsquark
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    Re: Predicting Total Movement after ( X ) Time

    Aha, i see

    This will return me ( by finding t ) time it takes for the Object to reach V_Target
    Vi * D^t + w * t <= V_Target ( where t is X/ unknown )

    By inserting the time i got from the previous equation ( and solving off cource ), i will recieve 'Distance Traveled To V_Target'
    ( Vi / Ln( D ) ) * ( D^t - 1 ) + ( 1 / 2 ) * ( w * t )^2

    Correct?
    I just want to make sure i got this right.

    How you manage to make equations out of what i descibe, i cannot fathom.
    Its kinda late for me, and i need to get some zzz.
    Ill try to implement this in the morning and post back on how it went down.

    In any case, awesome and thanks Ebaines!
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    MHF Contributor ebaines's Avatar
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    Re: Predicting Total Movement after ( X ) Time

    Quote Originally Posted by CakeSpear View Post
    Correct?
    I just want to make sure i got this right..... In any case, awesome and thanks Ebaines!
    Yes, that's corect, and you are most welcome!

    One additional thing to be aware of - the graph of distance versus velocity may not strictly speaking be a function, as there can be multiple values of distance for a given velocity. The attached figure shows graphs of distance versus time and velocity versus distance given the set of conditions V_i = 20, D= 0.9 and W = 0.5. Note that with tme the velocity first decreases (due to the D factor) but then starts to increase (due to W). The plot of D versus V shows that as velocity decreases D increases slowly, then as velocity starts to increase D increases faster. You would pick your target velocity off the horizontal axis on this second chart, then find the corresponding distance (or distances) at which this velocity occurs. Note that for any particular value of v_target there may be zero, one, or two solutions for distance.
    Attached Thumbnails Attached Thumbnails Predicting Total Movement after ( X ) Time-distance-velocity.jpg  
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    Re: Predicting Total Movement after ( X ) Time

    I thought it would be easy to rearrange this:
    Vi * D^t + w * t <= V_Target
    So that i could find/ isolate t

    But im lost...

    ...
    Im guessing there is no way of solving it ( finding t ) by rearranging the equation, and that by "numerica technique" you mean finding t by compairing to a graph...?
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  8. #8
    MHF Contributor ebaines's Avatar
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    Re: Predicting Total Movement after ( X ) Time

    There is no way to rearrange the equation  V_i D^t + wt = V_{target} to make time (t) the subject. By "numerical technique" what I meant is that there are several ways to get very close estimates of the value of t, and while such estimates are not exact they can be as accurate as you want. Newton's method is one such popular technique. If you are interested in determing t for f(t)= V_t, start by making an initial guess of what t might be, which we'll call t_1. For example you might guess that t=1 second. Put 1 in for t in the equation and calculate how far the result is from the desired value of V_target. That difference is the error of guess 1: E_1 = v(t_1)-V_Target. Next calculate the slope of the function at t=t_1, and use that slope to help determine the next guess t_2 using the realtionship:

    t_2 = t_1 - Error_1/slope(t_1)

    The slope can be calculated using the derivative of f(t) at t_1:

     Slope = f'(t_1) = V_i ( \ln D ) D^{t_1} + w

    Now you have your next guess t_2, and repeat the process until the value for Error is small enough to suit your needs. You should be able to iterate to a pretty good approximation in about 8 steps or so.

    However, be aware that as I pointed out previously and as shown on the graph that there may be two values for t that satisfy the equation, so you need to be sure to zero in on the specific one you want. Also if there are no values for t that satisfy the equation (which is quite possible depending on the value of V_target) the equations may blow up.
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