Physics 140: Another time, another place

When you try to fit real data with the Fundamental Kinematic Equation (FUNKE) you may run into an issue trying to estimate two parameters, x₀ and v₀, if the segment of data you're trying to fit starts a long time after t=0. That's because x₀ and v₀ are the position and velocity the cart would have had at t=0 if the acceleration had been constant all the way back to t=0. So these values may be a little weird. For example if you wait 5 seconds before starting the cart at the origin with a constant acceleration of +1 m/s/s, then x₀ would be 12.5 m and v₀ would be −5 m/s! — because that's where the cart would have had to start, and the velocity it would have had, if it had started 5 seconds earlier and travelled with a constant acceleration all the time.

Position vs time graph of a cart starting at x=0 at t=5s. The actual motion is in blue, the motion extrapolated back to t=0 is in red.

It might be easier to estimate the parameters if one expresses the FUNKE in a different way: use t₀, which is the time at which the parabola reaches its extreme value (either maximum or minimum) and x_p, the position at that time. In other words, the time and the position of the parabola's apex. Then the FUNKE becomes:

Your fitting parameters are x_p, t₀ and a, and a is the same as in the original form. Notice that v₀ is not in this equation because at the apex the velocity is zero.

You can do your modelling assignment with this equation. It is another correct way of expressing motion with constant acceleration. As long as you explain what you're doing you'll get full credit if it's right and partial credit if it's wrong and we understand what you're getting at.

There are many right ways of doing most problems. Unfortunately, there are many more wrong ways.

Can you find the mathematical relationship between x₀, v₀, a and x_p, t₀, a ?

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