Average Velocity and Instantaneous Velocity — Question 6

Question 6

A symmetric velocity estimate can still be biased

Let $u=t/(1\ \mathrm s)$ and $\vec r(t)=(u^3,2u)$ m. Estimate velocity at 2 s from the two measurements at $2\ \mathrm s-h$ and $2\ \mathrm s+h$, using their displacement divided by $2h$.

1. Find the exact instantaneous velocity at 2 s.

2. Derive the symmetric estimate in terms of $\delta=h/(1\ \mathrm s)$ and evaluate it for $h=0.5$ s.

3. How does the error change if h is halved? Does symmetry guarantee exactness for every trajectory?

Question 6 - Solution

A symmetric velocity estimate can still be biased

(a) Differentiate the cubic coordinate.

$$\vec v=\left(3u^2,2\right)\,\mathrm{m/s}\ \Rightarrow\ \boxed{\vec v(2\,\mathrm s)=(12,2)\,\mathrm{m/s}}$$ $\vec v=(3u^2,2)$ m/s, giving $(12,2)$ m/s at 2 s. The nonlinear x coordinate is the potential source of finite-interval error.

(b) Expand the endpoint difference.

$$\frac{(2+\delta)^3-(2-\delta)^3}{2\delta}=12+\delta^2,\quad\vec v_h=(12.25,2)\,\mathrm{m/s}$$ $[(2+\delta)^3-(2-\delta)^3]/(2\delta)=12+\delta^2$. Thus the estimate is $(12+\delta^2,2)$ m/s. For $h=0.5$ s, it is $(12.25,2)$ m/s, with x error $+0.25$ m/s.

(c) Use the error law rather than assuming cancellation.

$$\epsilon_x(h)=\delta^2\,\mathrm{m/s},\quad\epsilon_x(h/2)=\frac14\epsilon_x(h)=0.0625\,\mathrm{m/s}$$ Halving h divides $\delta^2$ by four, leaving x error 0.0625 m/s. Symmetry cancels some terms but not the cubic contribution. The estimate approaches the derivative as $h\to0$; finite symmetric measurements are not automatically exact.