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Straight-Line Motion: Connecting Position, Velocity, and Acceleration AB & BC

On a line, $v(t) = s'(t)$ and $a(t) = v'(t) = s''(t)$, so the particle is at rest where $v = 0$ rather than where $s = 0$, and acceleration is two differentiations away from position rather than one. Speed is $|v(t)|$: velocity carries direction in its sign, speed keeps only the size, so a velocity of $-8$ is a speed of 8.

Three errors run through the topic. Speed gets quoted with a negative sign, or averaged as though it were velocity. Whether the particle is speeding up gets decided from the sign of acceleration alone, when the real test is whether $v$ and $a$ share a sign: same sign speeds up, opposite signs slows down. And the three functions swap roles on a graph, so a velocity height is read as a position, or acceleration is found by differentiating position only once.

s(t) v(t) = s′(t) a(t) = v′(t) WHERE it is HOW FAST + WHICH WAY how the velocity changes d/dt d/dt a(t) = s″(t): TWO differentiations from position, not one AT REST means v(t) = 0 NOT s(t) = 0 parked at position 12: s = 12, v = 0 racing past the origin: s = 0, v = 9 The particle CHANGES DIRECTION only where v changes SIGN. Touching zero is not enough. Speed is |v(t)|, so it is never negative, and it is not what the sign of v is reporting.
Each arrow is one differentiation. Most errors here are a question asked of the wrong box.
THE TEST: COMPARE THE SIGN OF v WITH THE SIGN OF a a > 0 a < 0 v > 0 v < 0 SPEEDING UP SLOWING DOWN SLOWING DOWN SPEEDING UP same sign speed climbs opposite signs speed falls v = -6 a = -2, pushing the SAME way speed 6 → 8: SPEEDING UP, though a < 0 v = -6 a = +2, pushing BACK speed 6 → 4: SLOWING DOWN, though a > 0
Both bottom examples have the same velocity. The sign of the acceleration alone decides nothing without it.

The work

3 ways in · any order
Lesson
Straight-Line Motion: Connecting Position, Velocity, and Acceleration ›

Links position, velocity, and acceleration as successive derivatives, pins at rest to v = 0 rather than s = 0, separates speed from velocity, and settles speeding up and slowing down by comparing the signs of velocity and acceleration rather than reading acceleration alone.

Skill check · 10 scenarios
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Diagnostic
10-item topic check ›

Ten items spanning the three failure modes of this topic: treating speed and velocity as the same quantity, judging speeding up from the sign of acceleration alone, and swapping the roles of position, velocity, and acceleration when reading a graph or a formula.

Not yet available · 10 items
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Targeted Practice
Drill a single misconception ›

Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.

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