Various position functions from dx/dt

In summary, Calculus problems can have different forms of position functions, even when starting with v(t)=dx/dt. The function s(t) can be determined experimentally and can be found by using numerical techniques of differentiation. The equations of motion are based on experimentation over many years. The constants in the equations must have the correct units for the formula to work. The value of the coefficient of a variable can be 1 and is usually not written out.
  • #1
thedy
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Hi,I am just starting to solve calculus problems but,how is it possible,that from v(t) we can get so many differently position function.I mean,if we have v(t)=dx/dt we should always get just v=at+v0,and from this to get s(t)=1/2at^2+v0t+v0.But I have found many motion problems in which position functions have different forms,like for example:s(t)=t^2+4t^2 or s(t)=-t-t^2.And many others like if acceleration is not constant and so on.

So my question.If we solve some problem,we always start with v=dx/dt and therefore,we should get always just one,and still the same form of position function,should not we?

thanks for answers
 
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  • #2
thedy said:
I mean,if we have v(t)=dx/dt we should always get just v=at+v0
thanks for answers
Why should we always get that?
 
  • #3
I will ask otherwise.How do I get v(t) by derivation of s(t) if I do not know function s(t)?If I have some motion problem where we have function s(t) like known stuff,from what is this function determined?I mean,how I can know s(t) if I do not know v(t).And how I could know v(t) if I do not know s(t)?
If we are trying to solve some motion problem and we have just known variables and parameters,but we have no function,how we can solve the problem?I mean,it is determined,this function,which is neccessery only experimentally?
Thanks
 
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  • #4
arildno said:
Why should we always get that?
OK,well when we get other form?Not at+v0?If we still do dx/dy?I know,we must know x.What is hidden in ,,x".And that is my question.How we know,or where or from we can find out what is ,,x"?
Thanks
 
  • #5
thedy said:
I will ask otherwise.How do I get v(t) by derivation of s(t) if I do not know function s(t)?
If you don't know the formula for s(t), you can still get v(t) (or at least an approximation of it) by using numerical techniques of differentiation. For example, if you have these data points - (0, 2), (1, 5), (2, 10) - with the first coordinate being the time t and the second coordinate being position s, you can choose a polynomial or other function to fit the data to, and then differentiate. Alternatively, you can find the slope between successive pairs of points to get approximate values for v(t).
thedy said:
If I have some motion problem where we have function s(t) like known stuff,from what is this function determined?
Certain functions are reasonable models for projectile motion. The simpler functions such as s(t) = 16t2 give OK results and are easy to work with, but only take into account the gravitational force. For better results that take air resistance into account, the functions are more complicated, but give results that are more true to life. In general the equations of motion are arrived at by experimentation over hundreds of years (going back to Galileo).
thedy said:
I mean,how I can know s(t) if I do not know v(t).And how I could know v(t) if I do not know s(t)?
If we are trying to solve some motion problem and we have just known variables and parameters,but we have no function,how we can solve the problem?I mean,it is determined,this function,which is neccessery only experimentally?
Thanks

thedy said:
OK,well when we get other form?Not at+v0?If we still do dx/dy?I know,we must know x.What is hidden in ,,x".And that is my question.How we know,or where or from we can find out what is ,,x"?
Thanks
 
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  • #6
Thank you very much for answers. :)
 
  • #7
Hi,I just want to ensure.If I have for example velocity function:v(t)=2t-t^2,then t^2 expression is only time over 2 or some constant is next to t^2,but this constant=1?Or how is it possible,that we have only t^2,it seems,like we could have function that for example is s(t)=t^2 and then meter=second? Dimensionally seems to be incorrect...
THanks
 
  • #8
thedy said:
Hi,I just want to ensure.If I have for example velocity function:v(t)=2t-t^2,then t^2 expression is only time over 2 or some constant is next to t^2,but this constant=1?Or how is it possible,that we have only t^2,it seems,like we could have function that for example is s(t)=t^2 and then meter=second? Dimensionally seems to be incorrect...
THanks
The constants wouldn't be dimensionless. In your formula, v(t) = 2t - t2, the coefficient of t would have to be <distance>/<time2> and the coefficient of t2 would have to be <distance>/<time3>. That way the right side would be in units of <distance>/<time> to match the left side.

On the other hand, maybe the problem isn't concerned with units, in which case all they're interested in is the numeric value of the velocity.
 
  • #9
Mark44 said:
The constants wouldn't be dimensionless. In your formula, v(t) = 2t - t2, the coefficient of t would have to be <distance>/<time2> and the coefficient of t2 would have to be <distance>/<time3>. That way the right side would be in units of <distance>/<time> to match the left side.

On the other hand, maybe the problem isn't concerned with units, in which case all they're interested in is the numeric value of the velocity.
Yes,but what is the value of coefficient?For example the value of the coeffcient of t2 is 1?
 
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  • #10
thedy said:
Yes,but what is the value of coefficient?For example the value of the coeffcient of t2 is 1?
Yes. If the coefficient of some variable is 1, we usually don't bother to write it. For example x and 1x mean the same thing.
 

1. What is a position function?

A position function is a mathematical representation that describes the position of an object in relation to a fixed reference point. It is typically denoted as p(t) or s(t), where t represents time. It is used to track the location of an object as it moves over time.

2. How is a position function related to dx/dt?

Position functions are related to dx/dt through the derivative, which is represented by dx/dt. This represents the rate of change of the position function with respect to time. In other words, it shows how the position of an object is changing over time.

3. What does dx/dt tell us about an object's motion?

dx/dt tells us the velocity of an object, which is the rate of change of its position. It indicates the object's speed and direction of motion at any given time. A positive value for dx/dt indicates that the object is moving in the positive direction, while a negative value means it is moving in the negative direction.

4. How do we use dx/dt to find the acceleration of an object?

The acceleration of an object is the rate of change of its velocity, which can be found by taking the derivative of dx/dt. In other words, the second derivative of the position function is equal to the acceleration function, or a(t). This tells us how an object's velocity is changing over time.

5. Can we use position functions to predict an object's future position?

Yes, position functions can be used to predict an object's future position by plugging in a specific time value into the function. This will give us the position of the object at that time. However, this assumes that the object's motion remains constant and there are no external forces acting on it.

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