Q.Figure 2.14 gives the $x$-$t$ plot of a particle in one-dimensional motion. Three different equal intervals of time are shown. In which interval is the average speed greatest, and in which is it the least? Give the sign of average velocity for each interval.

Interval 3 is the steepest, so the average speed is greatest there; interval 2 is nearly flat near the maximum, so the average speed is least there. The particle moves toward $+x$ in intervals 1 and 2 (positive average velocity) and toward $-x$ in interval 3 (negat …

Concept

Over an interval,

For motion along a line without reversal within an interval, the average speed equals the magnitude of the average velocity, i.e. the magnitude of the chord's slope on the $x$-$t$ graph. The three intervals are equal in duration ($\Delta t$ the same), so comparing average speeds just means comparing how much $x$ changes.

Comparing the intervals

• Interval 1 (gentle rise): $x$ increases by a moderate amount, so a moderate positive slope. Average velocity $> 0$. …

Concept: Cross-Checking a Graph-Reading with an Explicit Numeric Model

Method: Build One Concrete, Consistent $x(t)$ and Compute the Three Average Velocities Directly

Comparing chord slopes "by eye" is the natural first pass, but it can be checked rigorously by constructing one explicit set of numbers consistent with the described shape (positive, rising gently to a rounded max, then falling steeply through zero) and simply computing each average velocity from the definition — the numbers below are only one example, but any numeric assignment matching the qualitative shape must give the same ranking, because that ranking follows from the shape, not the specific values chosen.

Steps

1. Assign consistent values at four equally spaced instants, matching "gentle rise, rounded near-flat maximum, then a steep fall through zero":

   $t$	0	2	4	6
   $x$	$0$	$8$	$9$	$-3$

   (Interval 1 $=[0,2]$: gentle rise. Interval 2 $=[2,4]$: near the rounded max, $x$ barely changes. Interval 3 $=[4,6]$: steep fall, crossing $x=0$.)

2. Compute the average velocity in each interval directly from $\bar v = \Delta x/\Delta t$:

3. Compare magnitudes (average speed): $|\bar v_2| = 0.5$ is the smallest, $|\bar v_3| = 6$ is the largest. This matches the qualitative reading: near the rounded top the curve is nearly flat (small $\Delta x$ over the interval), while the steep fall through zero produces the largest change in $x$ over an equal time.

4. Read the signs directly from the same table: $\bar v_1>0$, $\bar v_2>0$ (both are rises, even though interval 2's rise is tiny), $\bar v_3<0$ (a fall).

Why a numeric check is worth doing …