A-Level Maths (Edexcel 9MA0) · Mechanics Year 1 · Chapter 11: Variable acceleration · Lesson 3

Using integration and deriving the formulae

135 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 $\int (6 t^{2} - 4 t + 5) d t$ .
[2 marks]
Answer
A2
Needed today
Given that $f^{'} (t) = 4 t - 3$ and $f (2) = 7$ , find $f (t)$ .
[2 marks]
Answer
A3
Last lesson
A particle has velocity $v = t^{2} - 6 t + 8 m s^{- 1}$ for $0 \leq t \leq 5$ . Find its greatest speed.
[2 marks]
Answer
A4
Common slip
A particle has velocity $v = t - 3 m s^{- 1}$ . Mo says the distance travelled from $t = 0$ to $t = 4$ is $\int_{0}^{4} (t - 3) d t = - 4$ , so $4 m$ . Find the distance travelled.
[3 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 on the $x$ -axis with acceleration $a = (6 t - 4) m s^{- 2}$ at time $t$ seconds. When $t = 0$ it has velocity $3 m s^{- 1}$ and is at the point where $x = 2$ . Find $x$ when $t = 2$ .

Hint Integrate

a

to get

v

and use

v = 3

when

t = 0

. Then integrate

v

to get

x

and use

x = 2

when

t = 0

. Each integral needs its own constant.

B1
A particle moves on the $x$ -axis with acceleration $a = (12 - 6 t) m s^{- 2}$ at time $t$ seconds. When $t = 0$ it has velocity $- 9 m s^{- 1}$ and is at the point where $x = 5$ . Find the times at which it is at rest, its position at each of these times, and the distance between these positions.
[6 marks]
Answer

Example 2

A particle moves in a straight line with acceleration $a = (6 t + 4) m s^{- 2}$ at time $t$ seconds. Its displacement from $O$ is $6 m$ when $t = 1$ and $20 m$ when $t = 2$ . Find its displacement and velocity when $t = 3$ .

Hint You are not told the initial velocity, so keep both constants:

v = 3 t^{2} + 4 t + c

and

s = t^{3} + 2 t^{2} + c t + d

. The two positions give two equations in

c

and

d

B2
A particle moves in a straight line with acceleration $a = (12 t - 2) m s^{- 2}$ at time $t$ seconds. Its displacement from $O$ is $4 m$ when $t = 1$ and $17 m$ when $t = 2$ . Find its displacement and velocity when $t = 3$ .
[5 marks]
Answer

Example 3

A particle moves in a straight line with velocity $v = (8 t - 2 t^{2}) m s^{- 1}$ at time $t$ seconds. Find the distance it travels from $t = 0$ to $t = 6$ , and its displacement over this time.

Hint Find where

v = 0

in the interval. Integrate separately on each side of that time and add the sizes.

B3
A particle moves in a straight line with velocity $v = (t^{2} - 7 t + 10) m s^{- 1}$ at time $t$ seconds. Find the distance it travels between $t = 0$ and $t = 6$ .
[6 marks]
Answer

Example 4

A particle starts from rest at $O$ and moves in a straight line. Its velocity at time $t$ seconds is $v = 3 t^{2}$ for $0 \leq t \leq 2$ and $v = 16 - 2 t$ for $2 \leq t \leq 8$ . Find the total distance it travels.

Hint Check that the two expressions give the same velocity at

t = 2

. Integrate each over its own interval. Check whether

v

changes sign.

B4
A particle starts from rest at $O$ and moves in a straight line. Its velocity at time $t$ seconds is $v = 6 t - t^{2}$ for $0 \leq t \leq 4$ and $v = 12 - t$ for $4 \leq t \leq 12$ . Find the total distance it travels, and the acceleration of the particle just before and just after $t = 4$ .
[6 marks]
Answer

Example 5

A particle moves in a straight line with constant acceleration $a$ . At time $t = 0$ it is at $O$ with velocity $u$ . Show, by integration, that $v = u + a t$ and $s = u t + \frac{1}{2} a t^{2}$ .

Hint

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

with

a

a constant. Use the conditions at

t = 0

to find each constant.

B5
A particle moves in a straight line with constant acceleration $- 4 m s^{- 2}$ . At time $t = 0$ it is at $O$ with velocity $10 m s^{- 1}$ . Show by integration that $v = 10 - 4 t$ and $s = 10 t - 2 t^{2}$ , and hence show that $v^{2} = 100 - 8 s$ .
[6 marks]
Answer

Practice

A-Level

C1
A particle starts from rest at $O$ and moves in a straight line with acceleration $a = 2 t m s^{- 2}$ . Zoe finds its displacement at $t = 4$ like this: " $a = 8$ when $t = 4$ , so $s = u t + \frac{1}{2} a t^{2} = 0 + \frac{1}{2} (8) (16) = 64 m$ ." Explain why Zoe's method is wrong, and find the correct displacement.
[5 marks]
Answer

C2
A particle moves on the $x$ -axis with velocity $v = (3 t^{2} - 2) m s^{- 1}$ . When $t = 1$ it is at the point where $x = 5$ . Tia writes $x = t^{3} - 2 t$ and gives $x (3) = 21$ . Find the correct value of $x$ when $t = 3$ .
[5 marks]
Answer
C3
A particle moves in a straight line with velocity $v = (3 t^{2} - 15 t + 12) m s^{- 1}$ . Find $\int_{0}^{5} v d t$ and the distance travelled between $t = 0$ and $t = 5$ , and explain why they are different.
[6 marks]
Answer
C4
Two particles move in straight lines. The displacement of $P$ is $s = 5 + 12 t - 2 t^{2}$ metres and the displacement of $Q$ is $s = 12 t - 2 t^{3}$ metres, at time $t$ seconds. Show that $P$ moves with constant acceleration, stating its initial velocity and its acceleration. Explain why the constant acceleration formulae cannot be used for $Q$ .
[5 marks]
Answer
C5
A particle moves in a straight line with acceleration $a = (6 t - 8) m s^{- 2}$ . Its initial velocity is $5 m s^{- 1}$ . Find the time $T$ at which its velocity is next $5 m s^{- 1}$ , and its displacement from its starting point at time $T$ .
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
A particle $P$ moves on a straight line through $O$ . At time $t$ seconds its acceleration is $(8 - 2 t) m s^{- 2}$ . At $t = 0$ , $P$ is at $O$ with velocity $- 12 m s^{- 1}$ .
(a)
Find an expression for the velocity of $P$ at time $t$ .
[3 marks]
Answer
(b)
Find the times at which $P$ is at rest.
[2 marks]
Answer
(c)
Show that $P$ is back at $O$ when $t = 6$ .
[3 marks]
(d)
Find the total distance travelled by $P$ in the first $7$ seconds.
[4 marks]
Answer
Total for D1: 12 marks

D2
Kim says: " $v = 2$ at $t = 0$ and $v = 10$ at $t = 2$ , so the average velocity is $6 m s^{- 1}$ and the distance in the first $2$ seconds is $6 \times 2 = 12 m$ ."
A particle starts at $O$ and moves in a straight line with acceleration $3 t^{2} m s^{- 2}$ at time $t$ seconds. Its initial velocity is $2 m s^{- 1}$ .
(a)
Explain why Kim's method is not valid.
[2 marks]
(b)
Find the distance travelled in the first $2$ seconds.
[3 marks]
Answer
(c)
Find the time at which the velocity is $29 m s^{- 1}$ .
[2 marks]
Answer
(d)
Find the displacement from $O$ at that time.
[2 marks]
Answer
Total for D2: 9 marks
D3
A particle moves on the $x$ -axis. At time $t$ seconds its acceleration is $(2 t - 6) m s^{- 2}$ . At $t = 0$ the particle is at $O$ with velocity $10 m s^{- 1}$ in the positive $x$ -direction. Show that the particle never comes to rest, and find its displacement from $O$ when its speed is least.
[8 marks]
Answer
D4
A car starts from rest at a point $A$ and moves along a straight road. For $0 \leq t \leq 6$ its acceleration is $(2 - \frac{1}{3} t) m s^{- 2}$ . At $t = 6$ the brakes are applied and the car then slows down with constant deceleration $4 m s^{- 2}$ until it stops.
(a)
Find the speed of the car when $t = 6$ .
[3 marks]
Answer
(b)
Find the distance the car travels in the first $6$ seconds.
[3 marks]
Answer
(c)
Find the time at which the car stops and the total distance from $A$ to where it stops.
[3 marks]
Answer
(d)
Explain why the constant acceleration formulae may be used for the second stage but not for the first.
[2 marks]
Total for D4: 11 marks
D5
Two particles $P$ and $Q$ move along the same straight line. At time $t = 0$ both are at the point $O$ . $P$ starts from rest at $O$ , and its acceleration at time $t$ seconds is $(16 - 6 t) m s^{- 2}$ . $Q$ moves with constant velocity $16 m s^{- 1}$ in the positive direction.
(a)
Find expressions for the velocity of $P$ and the displacement of $P$ from $O$ at time $t$ .
[3 marks]
Answer
(b)
Find the times at which $P$ and $Q$ have the same velocity.
[3 marks]
Answer
(c)
Show that, for $t > 0$ , $P$ and $Q$ are level only when $t = 4$ . Hence explain why $P$ does not overtake $Q$ .
[4 marks]
Total for D5: 10 marks
D6
A particle $P$ moves in a straight line. At time $t$ seconds its acceleration is $(4 - 6 t) m s^{- 2}$ . When $t = 1$ , $P$ has velocity $5 m s^{- 1}$ . When $t = 2$ , $P$ is $10 m$ from a fixed point $O$ on the line, on the positive side.
(a)
Find the velocity of $P$ at time $t$ .
[3 marks]
Answer
(b)
Find the displacement of $P$ from $O$ at time $t$ .
[3 marks]
Answer
(c)
Find the time at which $P$ is at rest.
[2 marks]
Answer
(d)
Find the time at which $P$ is back at the point where it was when $t = 0$ .
[3 marks]
Answer
Total for D6: 11 marks

Extension

Stretch

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

E1
A particle starts from rest at $O$ and moves in a straight line. At time $t$ seconds its acceleration is $(2 t - k) m s^{- 2}$ , where $k$ is a positive constant. The particle is back at $O$ when $t = 6$ . Find $k$ , the greatest distance of the particle from $O$ for $0 \leq t \leq 6$ , and the total distance it travels in the first $8$ seconds.
[10 marks]
Hint 1 · What to try
Integrate twice. The particle starts from rest at $O$ , so both constants are zero. "Back at $O$ " means $s = 0$ .
Hint 2 · The first line
The furthest point from $O$ is where the particle turns round, where $v = 0$ . Check the end $t = 6$ too.
Hint 3 · The full method
For the distance in $8$ seconds, split at the time it turns round and add the sizes of the two moves.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can integrate to find velocity and displacement, using given conditions to find each constant.
I can find the distance travelled when a particle changes direction.
I can solve problems where the motion has two stages.
I can derive the constant acceleration formulae and explain when they cannot be used.

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