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Initial velocity by measuring negative acceleration?

  1. Feb 17, 2015 #1

    Quick Qustion : If an object is propelled through the air (e.g. a ball is thrown), once the ball has left the throwers hand it will immediatly start to have negative acceleration (at least in the x plane). If the negative acceleration is measured at the same point in time after the ball has left the throwers hand for each throw and all other variables being equal exept the velocity, is it possible to determine the initial velocity of the ball for each throw at the moment it left the throwers hand by using the measured negative acceleration only? If so, what method/formula is used to determine this?

    Many Thanks.
  2. jcsd
  3. Feb 17, 2015 #2


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    This depends on a lot of external factors, for example what you assume about the decelerating force, in particular its magnitude and how it relates to the velocity of the object.
  4. Feb 17, 2015 #3
    Thanks for your reply Orodruin. In response to you question, i am assuming that the object and all external factors are identical. The only variable is the initial velocity of the object. My question is not really object specific, more a question of can the initial velocity of an (any) object be determined if all you can measure is the the negative acceleration after the propelling force is no longer being applied.


  5. Feb 17, 2015 #4


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    As I said, this will depend on other external factors that you have not specified, such as if and how the negative acceleration depends on velocity and how well you can model this.
  6. Feb 17, 2015 #5
    I'm guessing you're thinking of the simplest case, where gravity is the only relevant force. In that case, it can't be done. The gravitational force is independent of the speed; you'll always measure a downward force of about (9.8 m/s^2)*(mass of ball), regardless of the speed. So the gravitational force gives you no information about the speed.

    With friction force, it's actually possible, though it's not necessarily trivial. The simple model of friction force for a falling/thrown object is:
    [tex] \vec{F}_{fric} = a \vec{v} + b \left|\vec{v}\right| \vec{v} [/tex]
    The friction force is directly related to the velocity, which means if you know the friction force, you should be able to solve for the velocity. The caveat is that you have to know the values of a and b, which would depend on the object and the air its moving through. Personally, I don't know how easy it is to find those values with sufficient accuracy.

    If there's no air resistance, I think you can actually use the rotation of the earth to figure out the velocity. There are forces other than gravity which show up due to the rotation of the earth (centrifugal force, coriolis force, etc) and if memory serves, some of these forces depend on the velocity of the object. So if you measured these rotational forces, you might be able to figure out the velocity (or at least get some information about it; I can't remember if you can solve for it completely). Of course, these forces are WAY smaller than the force of gravity, so measuring them would be a major challenge.

    To sum up, it's not impossible, but it's definitely not easy. Forces depending on velocity tend to be harder to deal with than forces depending on position.
  7. Feb 17, 2015 #6


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    If speed were the only variable and the deceleration were a function of that variable and is invertible (the inverse function exists) then yes, you would be able to determine it. Otherwise, you'd have a situation where velocity thrown is 10m/s and 20m/s respectively, while the deceleration would be equal for both. Are there any physical processes that could cause this? I'm not sure.
  8. Feb 18, 2015 #7
    If I understand correctly do you mean to say like you roll a ball across a perfectly smooth surface and the air or water resistance or friction is known. Therefore because the ball decelerates once it is rolled then the initial velocity can be calculated? Like at t = 0 velocity was 1 m/s but at t = 0.01 second velocity was reduced to .5 m/s. So it would be logical to assume that there was a faster speed at the beginning of the experiment because the velocity was slowed.
  9. Feb 18, 2015 #8
    Many thanks for your replies everyone.

    I'm am not thinking of a gravity only model. I am using a 'drag force' model :

    ## F_{D}=\frac{1}{2}\rho v^{2}AC_{D} ##

    The fixed values I am using are :

    ## \rho ## = 1.175799
    A = 0.001432m
    ## C_{D} ## = 0.5

    with v being variable.

    To simplify things, just looking at the x axis component (forward motion), i am using :

    ## F_{Dx}=-F_{D} \frac{v_{x}}{v} ##

    I am then using the differential velocity equation (again for x axis only) :

    ## \frac{dv_{x}}{dt}=a_{x}=\frac{F_{Dx}}{m} ##

    with m being fixed at 0.04592m

    Plugging all this into a Runge-Kutta 4th Order method and supplying the initial velocity allows me to see the acceleration at different points in time and also to plot positions.

    Then I perform the following test cases:

    Case 1: Set the CD to 0.5, Initial Velocity to 100m/s. The ball travels 70.99m after 1s. The acceleration values range from -91(rounded) at t=0 to -24(rounded) at t=1

    Case 2: Set the CD to 0.5, Initial Velocity to 50m/s, the ball travels 41.16m after 1s. and the acceleration values range from -23(rounded) at t=0 to -11(rounded) at t=1.

    What I want to know is if/how I can calculate that the initial velocity of (Case 1) is 100m/s and (Case 2) is 50m/s (assuming I did not know them) if all I know is that the acceleration at t=1 is -24 (Case 1) and -11 (Case 2)?

    Hope I have explained clearly. Please ask if not.


  10. Feb 19, 2015 #9
    Yes @ChineseSmeagol, basically this is correct but the ball is in not rolling. it is in free flight with only air resistance. The initial velocity (at t=0 is the maximum velocity) with it decreasing thereafter. See my previous post for details.

    Many Thanks,

  11. Feb 23, 2015 #10
    Many Thanks Mentallic. I think I may have solved this. I have taken the inverse of the Drag force equation :

    ##F_{D}=\frac{1}{2}\rho v^{2}AC_{D}## which gives:

    ##v=\pm \sqrt\frac{2 F_{D}}{\rho A C_{D}}##

    By working out what v is using this method at lets say t=1, I have then used the Runge Kutta method with v at t=0 as the initial velocity and using a negative time step starting from 1 and working back to 0 I get the initial velocity.

    Please let me know if you think there may be any flaws in my thinking.


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