Instantaneous Velocity and Speed — Question 4

Question 4

Zero velocity does not always mark a reversal

Two points move on separate east-positive straight tracks. In the formulas below, $t$ is numerical time in seconds and positions are numerical metres. Near $t=2$ their exact histories are $$x_A(t)=5+(t-2)^3,\qquad x_B(t)=5+(t-2)^2.$$ You may use $d[(t-c)^n]/dt=n(t-c)^{n-1}$. Instantaneous velocity is the position derivative and instantaneous speed its magnitude.

1. Find the position, velocity and speed of both points at 2 s.

2. Examine velocity signs immediately before and after 2 s. Which point reverses direction, and which continues in the same direction through a zero-velocity instant?

3. A student claims that every horizontal tangent is a maximum or minimum of position. Use these two histories to evaluate the claim.

4. Does either point remain at rest over a nonzero-duration interval around 2 s? Give a third position history that agrees with both points’ position and velocity at 2 s but remains at rest throughout $1\leq t\leq3\ \mathrm{s}$.

5. Explain why position and velocity at one instant cannot, by themselves, distinguish all three behaviors.

Question 4 - Solution

Zero velocity does not always mark a reversal

(a) Evaluate the derivative, not only the coordinate.

$$v_A(t)=3(t-2)^2\ \mathrm{m/s},\qquad v_B(t)=2(t-2)\ \mathrm{m/s}.$$ At 2 s both points have $\boxed{x=5\ \mathrm{m},\ v=0\ \mathrm{m/s},\ s=0\ \mathrm{m/s}}$.

(b) Reversal requires a change in direction across the instant.

For A, the square is positive on both sides of 2 s, so motion is eastward before and after, with an isolated zero velocity at 2 s. There is no reversal.

For B, the velocity is negative before 2 s and positive afterward. B approaches 5 m from greater positions, reaches its minimum, and reverses from westward to eastward motion.

(c) A zero slope is not sufficient for an extremum.

$$x_A(2+h)-5=h^3,\quad x_A(2-h)-5=-h^3;\qquad x_B(2\pm h)-5=h^2>0\quad(h>0).$$

A has a horizontal tangent at 2 s, but positions just before are below 5 m and positions just after are above 5 m. Thus it has neither a local maximum nor a local minimum there. B does have a minimum. The existence of a horizontal tangent alone does not decide whether a turning point occurs.

(d) Rest over an interval is a statement about every time in it.

Neither A nor B is constant on any interval around 2 s. Their speeds vanish only at the isolated instant. A suitable third history is $$\boxed{x_C(t)=5\ \mathrm{m}\quad\text{for all }1\leq t\leq3\ \mathrm{s}}.$$ Its velocity is zero throughout the interior of that interval.

(e) A single local value does not specify the surrounding history.

All three agree in position and velocity at 2 s. Yet A passes through that position in the same direction, B reverses there, and C remains there for an interval. Distinguishing these cases requires nearby-time information, such as the velocity signs or whether position stays constant, not merely the shared zero velocity.