One curve, four instruments. For $s(t)$ = position of a particle, match each question to the tool that answers it.

How fast at instant $t$ ?
When does it momentarily stop?
Net displacement over $[a,b]$ ?
Is it speeding up or slowing?

Derivative $s'(t)$
Compare signs of $s'$ and $s''$
$\int_a^b s'(t)\,dt$
Solve $s'(t) = 0$

Hints

Instantaneous questions call the derivative; accumulated questions call the integral.
"Speeding up" means velocity and acceleration push the same way, same sign.

Show the answer

How fast at instant $t$ ? → Derivative $s'(t)$
When does it momentarily stop? → Solve $s'(t) = 0$
Net displacement over $[a,b]$ ? → $\int_a^b s'(t)\,dt$
Is it speeding up or slowing? → Compare signs of $s'$ and $s''$

Why

A single motion problem exercises the whole toolkit: derivatives for instants, integrals for totals, sign analysis for behaviour. Fluency is knowing which tool the question is secretly asking for: the skill this unit was building.

Read the lesson: Integrated single-variable calculus problems →

Practise Integrated single-variable calculus problems

The app has 5 more questions on this lesson, and keeps your place in the course. Physics I is free to start.

Start Physics I free Get it onGoogle Play

One curve, four instruments. For s(t)s(t)s(t) = position of a particle, match each question to the tool that answers it.

Practise Integrated single-variable calculus problems

More questions on Integrated single-variable calculus problems

One curve, four instruments. For $s(t)$ = position of a particle, match each question to the tool that answers it.