Acceleration — Question 4

Question 4

Zero acceleration is not a complete description of motion

Three exact local velocity histories near $t=2\ \mathrm{s}$ are $$v_A(t)=4+(t-2)^3,\qquad v_B(t)=(t-2)^2,\qquad v_C(t)=2(t-2).$$ In these formulas $t$ is numerical seconds and velocities are numerical m/s. Differentiate using $d[(t-c)^n]/dt=n(t-c)^{n-1}$ to obtain acceleration in $\mathrm{m/s^2}$. Examine a sufficiently small neighborhood of 2 s.

1. Find velocity, speed and acceleration at 2 s for each history.

2. Which histories have a direction reversal at 2 s? Which are instantaneously at rest there? Does either condition require acceleration to vanish?

3. A student claims that zero acceleration always means a local maximum or minimum of velocity. Test this claim using A and B.

4. Does zero acceleration at one instant imply constant speed on a surrounding interval? Contrast A with a fourth history $v_D(t)=4\ \mathrm{m/s}$ throughout that interval.

5. Explain why the pair of readings $v=0,a=0$ at one instant does not establish that an object remains at rest for any duration.

Question 4 - Solution

Zero acceleration is not a complete description of motion

(a) Differentiate each history and evaluate at the same instant.

$$a_A=3(t-2)^2,\qquad a_B=2(t-2),\qquad a_C=2\quad(\mathrm{m/s^2}).$$

The accelerations are $a_A=3(t-2)^2$, $a_B=2(t-2)$ and $a_C=2$, in $\mathrm{m/s^2}$ under the stated numerical convention. $$\begin{aligned}(v_A,s_A,a_A)_{t=2}&=(4\ \mathrm{m/s},4\ \mathrm{m/s},0),\\(v_B,s_B,a_B)_{t=2}&=(0,0,0),\\(v_C,s_C,a_C)_{t=2}&=(0,0,2\ \mathrm{m/s^2}).\end{aligned}$$

(b) Test direction using nearby velocity signs.

A is eastward on both sides sufficiently close to 2 s. B is also eastward on both sides, with an isolated zero velocity at 2 s. C changes from westward to eastward and reverses. B and C are instantaneously at rest; only C reverses. C’s nonzero acceleration shows that neither instantaneous rest nor reversal requires zero acceleration.

(c) A zero slope need not create an extremum.

For A, velocity is below 4 m/s just before 2 s and above 4 m/s just after. It increases through a horizontal tangent, with neither a local maximum nor a local minimum. B does have a local minimum of velocity at 2 s. Both have zero acceleration there, so that single condition does not decide whether a velocity extremum occurs.

(d) An isolated zero derivative is weaker than a constant function.

A’s speed changes through 2 s even though its instantaneous acceleration there is zero. D instead has constant velocity and speed on the entire interval, with zero acceleration at every interior instant. At 2 s A and D have identical velocity and acceleration, but their neighboring histories differ.

(e) Two local readings still leave surrounding motion undetermined.

$$v_B(2)=a_B(2)=0,\qquad v_B(2+h)=h^2>0\quad(h\ne0).$$

B supplies a counterexample to the claim of sustained rest: $v_B(2)=a_B(2)=0$, but $v_B(t)>0$ for every nearby $t\ne2$. A history identically at rest would have the same pair of readings. Establishing rest over a duration requires information across that interval, not only values at one instant.