A-Level Maths (Edexcel 9MA0) · Mechanics Year 1 · Chapter 11: Variable acceleration · Lesson 2

Maxima and minima problems

128 marks

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Date

Do Now

Quick questions to start. Some check what you need today; some set a slip that costs marks.

A1
Needed today
Find the coordinates of the stationary points of $y = x^{3} - 6 x^{2} + 9 x + 2$ and state the nature of each.
[3 marks]
Answer
A2
Last lesson
A particle has displacement $s = t^{3} - 4 t^{2} + 5 t$ metres at time $t$ seconds. Find the times at which it is at rest.
[2 marks]
Answer
A3
Needed today
Write $2 t^{2} - 8 t + 11$ in the form $a (t - b)^{2} + c$ and state its least value.
[2 marks]
Answer
A4
Common slip
A particle has velocity $v = t^{2} - 6 t m s^{- 1}$ for $0 \leq t \leq 8$ . Sam says the greatest speed is $9 m s^{- 1}$ , because $\frac{d v}{d t} = 0$ when $t = 3$ and $v (3) = - 9$ . Find the greatest speed.
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

A particle moves in a straight line from a point $O$ . At time $t$ seconds its displacement from $O$ is $s$ metres, where $s = 12 t^{2} - 2 t^{3}$ , $t \geq 0$ . Find the greatest displacement of the particle from $O$ in the positive direction, showing that it is a maximum.

Hint At the greatest displacement the particle is instantaneously at rest, so

v = 0

. Use

a = \frac{d v}{d t}

to show it is a maximum.

B1
A particle moves in a straight line from a point $O$ . At time $t$ seconds its displacement from $O$ is $s$ metres, where $s = 15 t^{2} - t^{3}$ , $t \geq 0$ . Find the greatest displacement of the particle from $O$ in the positive direction, showing that it is a maximum.
[5 marks]
Answer

Example 2

A particle moves in a straight line. Its velocity at time $t$ seconds is $v m s^{- 1}$ , where $v = t^{2} - 8 t + 7$ , $0 \leq t \leq 6$ . Find the least velocity and the greatest speed of the particle in this interval.

Hint

a = 0

gives the least velocity. The greatest speed can be at that point (if

v

is negative there) or at an end of the interval. Compare the sizes.

B2
A particle moves in a straight line. Its velocity at time $t$ seconds is $v m s^{- 1}$ , where $v = 3 t^{2} - 18 t + 15$ , $0 \leq t \leq 7$ . Find the least velocity and the greatest speed of the particle in this interval.
[5 marks]
Answer

Example 3

A particle moves in a straight line with velocity $v m s^{- 1}$ at time $t$ seconds, where $v = 2 t^{2} - 6 t + 7$ , $t \geq 0$ . Find its least speed, and show that it is never at rest.

Hint Complete the square. A square is never negative.

B3
A particle moves in a straight line with velocity $v m s^{- 1}$ at time $t$ seconds, where $v = t^{2} - 5 t + 8$ , $t \geq 0$ . Find its least speed, and show that it is never at rest.
[4 marks]
Answer

Example 4

A drone moves vertically. Its height above the ground at time $t$ seconds is $h$ metres, where $h = \frac{t ^{4} - 12 t ^{3} + 36 t ^{2}}{10} + 20$ , $0 \leq t \leq 7$ . Find the greatest and least heights of the drone in this interval.

Hint Find where

\frac{d h}{d t} = 0

. Then compare

h

at those times AND at the two ends of the interval.

B4
A drone moves vertically. Its height above the ground at time $t$ seconds is $h$ metres, where $h = \frac{3 t ^{4} - 20 t ^{3} + 24 t ^{2}}{10} + 15$ , $0 \leq t \leq 6$ . Find the greatest and least heights of the drone in this interval.
[6 marks]
Answer

Example 5

A train leaves a station $A$ from rest and moves along a straight track towards a signal $400 m$ from $A$ . Its distance from $A$ at time $t$ seconds is modelled as $s = 0.6 t^{2} - 0.01 t^{3}$ metres. Show that, according to the model, the train stops before it reaches the signal. Find its greatest speed.

Hint The train stops when

v = 0

. Compare

s

at that time with

400

. The greatest speed is where

a = 0

B5
A cyclist starts from rest at $O$ and moves along a straight road. Her distance from $O$ at time $t$ seconds is $s = k t^{2} - 0.02 t^{3}$ metres, where $k$ is a constant. She comes to rest when she is $80 m$ from $O$ . Find the value of $k$ and her greatest speed.
[5 marks]
Answer

Practice

A-Level

C1
A particle moves in a straight line with velocity $v = 8 t - 2 t^{2} m s^{- 1}$ , $0 \leq t \leq 6$ . Leah says the greatest speed is $8 m s^{- 1}$ , because $\frac{d v}{d t} = 0$ at $t = 2$ . Find the greatest speed of the particle and explain Leah's mistake.
[5 marks]
Answer

C2
A particle moves on a straight line through $O$ . Its displacement from $O$ is $s = t^{3} - 12 t^{2} + 21 t$ metres for $0 \leq t \leq 8$ . Omar says the greatest distance from $O$ is $10 m$ , at $t = 1$ . Find the greatest distance of the particle from $O$ .
[5 marks]
Answer
C3
A particle moves in a straight line with velocity $v = - 3 t^{2} + 18 t + 21 m s^{- 1}$ , $t \geq 0$ . Find the greatest velocity of the particle, showing that it is a maximum, and find the time at which the particle comes to rest.
[6 marks]
Answer
C4
A ball is thrown upwards. Its height above the ground at time $t$ seconds is modelled as $h = 4.5 t^{2} - 0.5 t^{3}$ metres. Tom says: "The greatest height is when the acceleration is zero, so $9 - 3 t = 0$ , $t = 3$ and the greatest height is $27 m$ ." Explain Tom's mistake and find the greatest height.
[5 marks]
Answer
C5
A particle moves on the $x$ -axis with displacement $x = 2 t^{3} - 0.25 t^{4}$ metres at time $t$ seconds, $t \geq 0$ . Find the greatest velocity of the particle and the greatest acceleration of the particle.
[6 marks]
Answer

Exam-style questions

Edexcel exam style

D1
A train leaves a station $A$ from rest and moves along a straight track to the next station $B$ , where it stops. Until it stops at $B$ , its distance from $A$ at time $t$ seconds is modelled as $s$ metres, where $s = 0.9 t^{2} - 0.005 t^{3}$ .
(a)
Find the distance from $A$ to $B$ .
[3 marks]
Answer
(b)
Find the greatest speed of the train.
[3 marks]
Answer
(c)
The speed limit on this track is $45 m s^{- 1}$ . Find, according to the model, the length of time for which the train is travelling faster than the speed limit.
[4 marks]
Answer
(d)
Explain why the model is not valid for $t > 120$ .
[2 marks]
Total for D1: 12 marks

D2
Ria finds the least speed like this: " $a = 6 t - 24 = 0$ at $t = 4$ , and $v (4) = - 12$ , so the least speed of $P$ is $12 m s^{- 1}$ ."
A particle $P$ moves in a straight line. At time $t$ seconds its velocity is $v m s^{- 1}$ , where $v = 3 t^{2} - 24 t + 36$ , $0 \leq t \leq 5$ .
(a)
Explain the mistake in Ria's working.
[2 marks]
(b)
Find the greatest speed of $P$ .
[3 marks]
Answer
(c)
Find all the times at which the speed of $P$ is $9 m s^{- 1}$ .
[4 marks]
Answer
Total for D2: 9 marks
D3
A particle $P$ moves on the $x$ -axis. At time $t$ seconds its displacement from $O$ is $x$ metres, where $x = t^{3} - 12 t^{2} + 45 t$ , $0 \leq t \leq 6$ . The point $B$ on the $x$ -axis has $x = 60$ . Find the least distance between $P$ and $B$ during the motion, and the times at which it occurs.
[8 marks]
Answer
D4
A particle $P$ moves on a straight line through $O$ . At time $t$ seconds its displacement from $O$ is $s$ metres, where $s = p t^{3} + q t^{2} + 12 t$ and $p$ and $q$ are constants. $P$ is instantaneously at rest when $t = 2$ and when $t = 3$ .
(a)
Find the values of $p$ and $q$ .
[4 marks]
Answer
(b)
Find the displacement of $P$ at each time it is at rest, and state whether each is a greatest or a least displacement near that time.
[3 marks]
Answer
(c)
Find the least velocity of $P$ .
[2 marks]
Answer
Total for D4: 9 marks
D5
A lift moves vertically upwards from rest at the ground floor. At time $t$ seconds its velocity is $v m s^{- 1}$ , where $v = 3 t - 0.3 t^{2}$ , $0 \leq t \leq 10$ . A passenger of mass $60 kg$ stands on the floor of the lift.
(a)
Show that the lift is at rest when $t = 10$ .
[2 marks]
Answer
(b)
Find the greatest speed of the lift.
[3 marks]
Answer
(c)
The height of the lift above the ground floor is $h = 1.5 t^{2} - 0.1 t^{3}$ metres. Show that this agrees with the velocity, and find the height through which the lift rises in the $10$ seconds.
[2 marks]
Answer
(d)
Find the greatest acceleration of the lift and the force exerted by the floor on the passenger at that instant.
[3 marks]
Answer
Total for D5: 10 marks
D6
A particle $P$ moves on the $x$ -axis. At time $t$ seconds its displacement from $O$ is $x$ metres, where $x = t^{3} - 3 t^{2} - 9 t + 4$ , $0 \leq t \leq 5$ .
(a)
Find the least value of $x$ in this interval.
[3 marks]
Answer
(b)
Find the greatest distance of $P$ from $O$ in this interval.
[3 marks]
Answer
(c)
Find the greatest speed of $P$ in this interval.
[3 marks]
Answer
Total for D6: 9 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A car starts from rest at a point $A$ and moves along a straight road. Its distance from $A$ at time $t$ seconds is modelled as $s = p t^{2} - q t^{3}$ metres, where $p$ and $q$ are positive constants. The car comes to rest at a point $B$ , $300 m$ from $A$ , when $t = 20$ . Find $p$ and $q$ , the greatest speed of the car, and the length of time for which its speed is greater than $18 m s^{- 1}$ .
[10 marks]
Hint 1 · What to try
Write down $v$ . "Comes to rest at $B$ when $t = 20$ " gives two equations: $v (20) = 0$ and $s (20) = 300$ .
Hint 2 · The first line
The greatest speed is where $\frac{d v}{d t} = 0$ . Check that $v$ is positive there.
Hint 3 · The full method
Solve $v > 18$ as a quadratic inequality. The length of time is the difference between its two roots.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the greatest displacement or velocity by setting the derivative to zero and checking its nature.
I can find the greatest speed on an interval, checking both directions and the end points.
I can find a least speed by completing the square and show a particle is never at rest.
I can use a given greatest value to find a constant in a model.

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