Greatest or least values in motion

Greatest or least velocity, speed, acceleration or displacement on an interval

In 9 of 20 recent exams
Paper 1 or 2
1 to 5 marks

How it is set in the exam

Paper 1 (no calculator) (3 exam versions) and Paper 2 (calculator) (6), 1–5 marks per part, often two or three linked parts. Common (9 of 20 exam versions). In 4 of them the answer is at an end of the interval, and checking the end has a mark of its own.

Spot it in the exam

Exam phrases to spot (4)

“find the maximum velocity”
“find the greatest speed of the particle for $0 \le t \le 5$ ”
“find the minimum acceleration”
“find the maximum displacement”

In disguise (5)

"the maximum distance of the particle from $O$ ": the greatest $\lvert s \rvert$ , so a negative $s$ can win
the greatest acceleration when $a(t)$ is one sine or cosine with no constant term, such as $a = -18\sin 3t$ (from $s = 1 + 2\sin 3t$ ): its amplitude, $18$ m s $^{-2}$ , if a peak of $a$ is in the interval (an end counts). Otherwise use $\frac{da}{dt} = 0$ and the ends: with a constant term ( $a = 2 + 3\cos t$ peaks at $5$ , not $3$ ), a growing factor (as in Example 2.2) or no peak inside the interval, there is no shortcut
$s(t)$ is given and the greatest velocity is asked for: the greatest gradient of $s$ , which can be at the start
"the least speed": $0$ whenever $v = 0$ somewhere in the interval
the greatest distance between two particles: the gap $\lvert s_B - s_A \rvert$ , with candidates where $v_A = v_B$ , and the ends

Looks like it, but isn't (4)

"when is the acceleration zero?", with no greatest or least asked for: Velocity and acceleration: at rest, direction, speed
the greatest rate of change of a quantity that isn't a motion (a population, a volume): Concavity and points of inflexion
the greatest value of a quantity that isn't a motion, on a given interval: Stationary points: find them and justify their nature, comparing the ends
the velocity is a sinusoid and no derivative is asked for: the times of its extremes, and how many there are, come from the period (Read a sinusoid's features)

Worked example

The greatest speed is at the start

Paper 1 (no calculator)

Question

A particle moves in a straight line with velocity $v(t) = 2t^3 - 21t^2 + 60t - 30$ m s $^{-1}$ , for $0 \le t \le 6$ .
(a) Find the acceleration $a(t)$ . [2]
(b) Find the greatest velocity of the particle. [4]
(c) Find the greatest speed of the particle. [3]

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See it move: The ends are candidates too

$a(t) = v'(t) = 6t^{2} - 42t + 60 = 6(t - 2)(t - 5)$ , so $a = 0$ at $t = 2$ and $t = 5$ .
The candidates on $0 \le t \le 6$ : $v(0) = -30$ (an end), $v(2) = 22$ ( $a = 0$ ), $v(5) = -5$ ( $a = 0$ ), $v(6) = 6$ (an end). Where $a = 0$ outside the interval doesn't count; the ends always do.
The greatest velocity is $22$ m s $^{-1}$ , at $t = 2$ ( $a = 0$ ); the least is $-30$ , at $t = 0$ (an end).
Speed is $\lvert v \rvert$ : the sizes $30$ , $22$ , $5$ , $6$ . The greatest speed is $30$ m s $^{-1}$ , at $t = 0$ (an end). $v$ passes through 0 on the way, so the least speed is 0.
(b) The greatest velocity is $22$ m s $^{-1}$ , at $t = 2$ . (c) The greatest speed is $30$ m s $^{-1}$ , at the start: a large negative velocity is the least velocity but can be the greatest speed.

Steps

Name the quantity.

It decides both the candidates and the comparison:

velocity $v$
Candidates inside the interval
$a = 0$
Also
both ends
Compare
values, with their signs
speed $\lvert v \rvert$
Candidates inside the interval
$a = 0$
Also
both ends
Compare
sizes: a negative $v$ can win. The least speed is $0$ whenever $v = 0$ somewhere in the interval
acceleration $a$
Candidates inside the interval
$\frac{da}{dt} = 0$ (for one sine or cosine with no constant term: its amplitude, if a peak is in the interval)
Also
both ends
Compare
values
displacement $s$
Candidates inside the interval
$v = 0$ : where $v$ changes from $+$ to $-$ (greatest) or from $-$ to $+$ (least)
Also
both ends
Compare
values
distance from $O$ , $\lvert s \rvert$
Candidates inside the interval
$v = 0$
Also
both ends
Compare
sizes: a negative $s$ can win. The least distance from $O$ is $0$ whenever $s = 0$ somewhere in the interval

Greatest or least …	Candidates inside the interval	Also	Compare
velocity $v$	$a = 0$	both ends	values, with their signs
speed $\lvert v \rvert$	$a = 0$	both ends	sizes: a negative $v$ can win. The least speed is $0$ whenever $v = 0$ somewhere in the interval
acceleration $a$	$\frac{da}{dt} = 0$ (for one sine or cosine with no constant term: its amplitude, if a peak is in the interval)	both ends	values
displacement $s$	$v = 0$ : where $v$ changes from $+$ to $-$ (greatest) or from $-$ to $+$ (least)	both ends	values
distance from $O$ , $\lvert s \rvert$	$v = 0$	both ends	sizes: a negative $s$ can win. The least distance from $O$ is $0$ whenever $s = 0$ somewhere in the interval

Find the candidates:
by solving; with maximum or minimum of a graph, on the graph of the quantity itself. If that quantity is a derivative of what you were given ( $v$ from $s$ , or $a$ from $v$ ), draw it with graph of the derivative, or differentiate by hand first.
Values:
substitute (), or read them from the GDC (). A displacement at a candidate time $t^*$ is $s(t^*) = s(0) + \int_0^{t^*} v\,dt$ (definite integral), with $t^*$ at full accuracy (keep full-accuracy values).
Compare every candidate, the ends included.
No nature test is needed: the comparison decides.
Answer:
the value with its units, and the time if it is asked for.

Common mistakes

Comparing the turning points only
the end of the interval can win, and checking it has its own mark (Examples 2.1 and 2.2).
The greatest speed from the greatest velocity
a large negative velocity is a large speed.
The least speed as the least velocity
the least speed is $0$ whenever $v = 0$ somewhere in the interval.

More mistakes (3)

The greatest displacement where $v$ is greatest
it is where $v = 0$ (changing from $+$ to $-$ ), or at an end.
Displacement for distance from $O$
the distance from $O$ is $\lvert s \rvert$ (Example 2.3(c)).
Degree mode
with trig: ().

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