Average Acceleration and Instantaneous Acceleration — Question 8

Question 8

Velocity samples give an average, not an exact local acceleration

At times 1, 2 and 3 s, measured velocity vectors are $(1,4)$, $(4,4)$ and $(9,4)$ m/s. Assume values are exact samples, with no stated interpolation model.

1. Find average acceleration over each adjacent interval and over 1 to 3 s.

2. If the model $\vec v(t)=((t/\mathrm s)^2,4)$ m/s is additionally assumed, find acceleration at 2 s.

3. Can the three samples alone establish that same instantaneous acceleration? Construct a velocity modification that preserves the samples but changes the derivative at 2 s.

Question 8 - Solution

Velocity samples give an average, not an exact local acceleration

(a) Compute finite differences.

$$\boxed{\vec a_{12}=(3,0),\quad\vec a_{23}=(5,0),\quad\vec a_{13}=(4,0)}\quad\mathrm{m/s^2}$$ The averages are $(3,0)$ m/s$^2$ from 1 to 2 s, $(5,0)$ m/s$^2$ from 2 to 3 s, and $(4,0)$ m/s$^2$ over the full 2 s interval. These are determined by the samples.

(b) Differentiate the assumed interpolation.

$$\vec a(t)=\frac{d(u^2,4)}{dt}\,\mathrm{m/s}=(2u,0)\,\mathrm{m/s^2},\quad\vec a(2)=(4,0)\,\mathrm{m/s^2}$$ The quadratic model gives $\vec a=(2t/\mathrm s,0)$ m/s$^2$, hence $(4,0)$ m/s$^2$ at 2 s. Its agreement with the symmetric average follows from the model, not from a general property of three measurements.

(c) Exhibit a different smooth history.

$$f(u)=c(u-1)(u-2)(u-3),\quad f(1)=f(2)=f(3)=0,\quad f'(2)=-c$$ Write $u=t/\mathrm s$ and add $c(u-1)(u-2)(u-3)$ m/s to $v_x$, with dimensionless c. It vanishes at all sample times, but its derivative at $u=2$ is $-c$ m/s$^2$. Thus the local x acceleration becomes $4-c$ m/s$^2$ while all three samples remain unchanged.