Finding the Velocity graph.

 

To find the velocity from a curved position time graph you must find the the slope of the position time graph at 6 places along the curve.

 

Unfortunately your position graph may not be accurate enough to do this.  So you must make a new more accurate position graph.  This can be done by using the equation you found from the position time graph.

  1. First make sure your linear test graph is well drawn for the line of best fit.
  2. On good graph paper you should be able to get 2 to 3 digits for the slope remembering to estimate the last digit.
  3. Because the graphing procedure averages over the data points the equation should be more accurate than the original data.
  4. Now using the equation plug in values for t from 0 to 5.0 in half second steps (0.0, 0.5, 1.0…) and make a table of the more accurate value.
  5. Using this table make a new X-t graph with the data spread out as much as possible.
  6. Draw a very smooth curve through the data points.
  7. Now find the velocity at 5 to 6 points by taking the slope of the tangent line at 5 to 6 places along the curve.  This procedure is shown in pages R26-R27 of Minds on Physics.
  8. Notice that at zero time you are at the bottom of the half parabola.  So the tangent line at that point is horizontal.
  9. You should now have 5 or 6 velocities calculated at different times.
  10. Now make the final graph of velocity vs time and find the equation.

 

At this point you have 4 graphs.

  1. The original X vs t using the raw data
  2. The test graph used to get the equation X=…
  3. The new accurate calculated X vs t  with the tangent lines used to find the velocities
  4. The velocity vs time graph

 

You also have two equations.

  1. The position vs time equation X= mX t²
  2. The velocity vs time equation V= mV t

 

The coefficient mX in the position equation does not have the same value as the coefficient mV in the velocity equation.  How many times greater is one than the other???

 

Remember the idea that you can find the acceleration between two times by finding the slope of a straight line drawn between the two times or by V=ΔX/ Δt.  Also remember that Δ means change in.  When you are doing this you are finding the between two times.  So we call it average velocity.

 

But when you find the slope of the tangent line at one place on an X-t graph, you are finding the velocity at only that time.  This is a velocity at only that instant in time.  So we call it instantaneous velocity.