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

More constant acceleration problems

122 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/a-level-mechanics-more-constant-acceleration-problems/

Date

Do Now

Quick questions to start. They use what you need today.

A1
Needed today
A car accelerates uniformly from $6 m s^{- 1}$ to $20 m s^{- 1}$ in $7 s$ . Find its acceleration and the distance it travels in this time.
[3 marks]
Answer
A2
Needed today
Solve $2 t^{2} - 11 t + 12 = 0$ .
[2 marks]
Answer
A3
Common slip
A train moves at $54 km h^{- 1}$ . It slows down uniformly and stops in $25 s$ . Find its deceleration.
[2 marks]
Answer
A4
Common slip
A cyclist starts from rest and accelerates uniformly to $8 m s^{- 1}$ in $4 s$ , then rides at $8 m s^{- 1}$ for $6 s$ . Use a velocity–time graph to find the distance she travels.
[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

Starting from $v = u + a t$ and $s = \frac{1}{2} (u + v) t$ , show that $v^{2} = u^{2} + 2 a s$ . A particle moving in a straight line with constant acceleration passes a point $A$ with speed $5 m s^{- 1}$ and a point $B$ with speed $13 m s^{- 1}$ , where $A B = 36 m$ . Find its acceleration.

Hint To remove

t

, make

t

the subject of

v = u + a t

and put it into the other formula. In the problem

t

is not given and not asked for, so use the formula with no

t

B1
Starting from $v = u + a t$ and $s = \frac{1}{2} (u + v) t$ , show that $s = v t - \frac{1}{2} a t^{2}$ . A car moves in a straight line with constant deceleration $1.5 m s^{- 2}$ . In $8 s$ it travels $72 m$ . Find its speed at the end and at the start of the $8 s$ .
[6 marks]
Answer

Example 2

A particle moves along a straight line with constant deceleration $2 m s^{- 2}$ . At $t = 0$ it passes a point $O$ with speed $11 m s^{- 1}$ , moving towards a point $A$ , where $O A = 24 m$ . Find the times when the particle is at $A$ , its velocity each time, and the time when it returns to $O$ .

Hint Take the direction from

O

A

as positive, so

a = - 2

. You know

u

a

and

s

and want

t

: use

s = u t + \frac{1}{2} a t^{2}

. This gives a quadratic, and both roots can be correct.

B2
A particle moves along a straight line with constant deceleration $3 m s^{- 2}$ . At $t = 0$ it passes a point $O$ with speed $16 m s^{- 1}$ , moving towards a point $A$ , where $O A = 40 m$ . Find the times when it is at $A$ , the length of time for which it is beyond $A$ , and the time when it returns to $O$ .
[5 marks]
Answer

Example 3

A particle $P$ moves on the $x$ -axis with constant acceleration $- 3 m s^{- 2}$ . At $t = 0$ it passes through the origin $O$ moving in the positive $x$ -direction with speed $12 m s^{- 1}$ . Find the position of $P$ at $t = 10$ , its velocity then, and the total distance it travels in the $10 s$ .

Hint

s = u t + \frac{1}{2} a t^{2}

gives the displacement, not the distance. Find when

P

stops (

v = 0

): before then it moves forward, after then it comes back.

B3
A particle moves on the $x$ -axis with constant acceleration $- 2.5 m s^{- 2}$ . At $t = 0$ it passes through $O$ moving in the positive $x$ -direction with speed $15 m s^{- 1}$ . Find its position at $t = 14$ , its velocity then, and the total distance it travels in the $14 s$ .
[5 marks]
Answer

Example 4

A cyclist rides along a straight road at a constant $9 m s^{- 1}$ . At $t = 0$ she passes a point $O$ . At $t = 4$ a motorbike starts from rest at $O$ and accelerates at $2 m s^{- 2}$ along the road in the same direction. Find the time when the motorbike draws level with the cyclist and the distance from $O$ at which this happens.

Hint Write both displacements from

O

in terms of

t

, the time since the cyclist passed

O

. The motorbike has been moving for

t - 4

seconds. They are level when the displacements are equal. The motorbike has not started for

t < 4

B4
A cyclist rides along a straight road at a constant $6 m s^{- 1}$ . At $t = 0$ he passes a point $O$ . At $t = 3$ a motorbike starts from rest at $O$ and accelerates at $3 m s^{- 2}$ in the same direction. Find the time when the motorbike draws level with the cyclist, the distance from $O$ , and the speed of the motorbike then.
[5 marks]
Answer

Example 5

A particle moves in a straight line with constant acceleration. It travels $14 m$ in the 4th second of its motion and $20 m$ in the 6th second. Find its initial speed and its acceleration.

Hint The 4th second runs from

t = 3

t = 4

. The distance in it is

s (4) - s (3)

, where

s (t) = u t + \frac{1}{2} a t^{2}

B5
A particle moves in a straight line with constant acceleration. It travels $11 m$ in the 3rd second of its motion and $21 m$ in the 8th second. Find its initial speed, its acceleration, and the distance it travels in the first $10 s$ .
[5 marks]
Answer

Practice

A-Level

C1
A cyclist starts from rest and accelerates uniformly at $1.5 m s^{- 2}$ for $8 s$ . She then decelerates uniformly at $0.5 m s^{- 2}$ . Ali finds her distance from the start after $12 s$ like this: " $s = 0 \times 12 + \frac{1}{2} \times 1.5 \times 1 2^{2} = 108$ m". Explain Ali's mistake and find the correct distance.
[5 marks]
Answer

C2
Car $A$ moves along a straight road at a constant $16 m s^{- 1}$ . At the instant it passes a point $O$ , car $B$ starts from rest at $O$ and accelerates at $2 m s^{- 2}$ in the same direction. Sam says: " $B$ catches $A$ when their speeds are equal, at $t = 8$ ." Explain why Sam is wrong. Find the time when $B$ catches $A$ , and the greatest distance between the cars before then.
[5 marks]
Answer
C3
A particle moves with constant acceleration along a straight line through $A$ , $M$ and $C$ , where $M$ is the midpoint of $A C$ . Its speed is $4 m s^{- 1}$ at $A$ and $16 m s^{- 1}$ at $C$ . Leo says its speed at $M$ is $\frac{4 + 16}{2} = 10 m s^{- 1}$ . Find the speed at $M$ , and explain what $10 m s^{- 1}$ is.
[5 marks]
Answer
C4
A particle moves on the $x$ -axis with constant acceleration $- 2 m s^{- 2}$ . At $t = 0$ it passes through $O$ moving in the positive $x$ -direction with speed $6 m s^{- 1}$ . The point $B$ is on the axis at $x = - 16$ . Jo writes " $v^{2} = 6^{2} + 2 (- 2) (16) = - 28$ , which is impossible, so the particle never reaches $B$ ". Explain her mistake. Find the velocity of the particle and the time when it reaches $B$ .
[5 marks]
Answer
C5
Two points $A$ and $B$ are $84 m$ apart on a straight track. At $t = 0$ particle $P$ starts from rest at $A$ and moves towards $B$ with acceleration $2 m s^{- 2}$ . At the same instant particle $Q$ passes $B$ moving towards $A$ with speed $5 m s^{- 1}$ and acceleration $1 m s^{- 2}$ . Find when and where the particles meet, and the speed of each then.
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
A train moves along a straight track through three points $P$ , $Q$ and $R$ , in that order. It passes $P$ with speed $u m s^{- 1}$ and accelerates uniformly at $0.4 m s^{- 2}$ to $Q$ , where $P Q = 640 m$ and its speed is $3 u m s^{- 1}$ . From $Q$ it decelerates uniformly and comes to rest at $R$ , where $QR = 960 m$ .
(a)
Show that $u = 8$ .
[3 marks]
Answer
(b)
Find the time taken to travel from $P$ to $R$ .
[3 marks]
Answer
(c)
Find the distance of the train from $P$ when $t = 60$ , where $t$ seconds is the time since it passed $P$ .
[3 marks]
Answer
(d)
Find the value of $t$ when the train is $1000 m$ from $P$ .
[3 marks]
Answer
Total for D1: 12 marks

D2
A particle moves on the $x$ -axis with constant acceleration $- 3.5 m s^{- 2}$ . At $t = 0$ it passes through $O$ moving in the positive $x$ -direction with speed $14 m s^{- 1}$ . Rui finds the distance travelled by the particle in the first $6 s$ : " $s = u t + \frac{1}{2} a t^{2} = 14 \times 6 - \frac{1}{2} \times 3.5 \times 36 = 21$ , so the distance is $21$ m."
Rui is wrong.
(a)
Explain Rui's mistake.
[2 marks]
(b)
Find the distance travelled in the first $6 s$ .
[4 marks]
Answer
(c)
Find all the times when the particle is $21 m$ from $O$ .
[3 marks]
Answer
Total for D2: 9 marks
D3
A particle $P$ moves along a straight line with constant deceleration $a m s^{- 2}$ . At time $t = 0$ it passes a point $O$ with speed $10 m s^{- 1}$ , moving towards a point $A$ , where $O A = 21 m$ . $P$ passes $A$ , comes to instantaneous rest, and passes $A$ again $4 s$ after it first passed $A$ . Find the value of $a$ and the greatest distance of $P$ from $O$ .
[8 marks]
Answer
D4
A car moves along a straight horizontal road at $u m s^{- 1}$ when the driver sees an obstacle $60 m$ ahead. In a model, the car keeps moving at $u m s^{- 1}$ for $0.7 s$ , the driver's reaction time, and then decelerates uniformly at $6 m s^{- 2}$ until it stops.
(a)
Given that $u = 20$ , show that the car stops before it reaches the obstacle, and find how far from the obstacle it stops.
[3 marks]
Answer
(b)
Show that the car stops before it reaches the obstacle when $u^{2} + 8.4 u - 720 < 0$ .
[3 marks]
Answer
(c)
Hence find the greatest speed at which the car can be moving if it is to stop before the obstacle. Give your answer in $km h^{- 1}$ .
[3 marks]
Answer
(d)
On another day the car is moving at $25 m s^{- 1}$ when the driver sees the obstacle $60 m$ ahead. Find the speed of the car when it hits the obstacle.
[3 marks]
Answer
(e)
State one limitation of the model, and say how it would change your answer to part (c).
[2 marks]
Total for D4: 14 marks
D5
A particle moves along a straight line with constant acceleration $a m s^{- 2}$ . Its initial speed is $u m s^{- 1}$ .
(a)
Starting from $v = u + a t$ and $s = \frac{1}{2} (u + v) t$ , show that $s = u t + \frac{1}{2} a t^{2}$ .
[3 marks]
Answer
(b)
Hence show that the distance travelled in the $n$ th second is $u + \frac{1}{2} (2 n - 1) a$ .
[3 marks]
Answer
(c)
The distances travelled in the 3rd second and the 7th second are in the ratio $1 : 2$ . The particle travels $18 m$ in the 5th second. Find $u$ and $a$ .
[4 marks]
Answer
Total for D5: 10 marks

Extension

Stretch

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

E1
Car $A$ moves along a straight road with constant acceleration $1 m s^{- 2}$ . At $t = 0$ it passes a point $O$ with speed $10 m s^{- 1}$ . Car $B$ passes $O$ , in the same direction, $4 s$ later, with speed $30 m s^{- 1}$ and a constant deceleration of $1 m s^{- 2}$ . Show that $B$ draws level with $A$ twice. Find the two times, the distance from $O$ of each meeting point, and the greatest distance by which $B$ is ahead of $A$ .
[9 marks]
Hint 1 · What to try
Measure all times from when $A$ passes $O$ . $B$ has been moving past $O$ for $t - 4$ seconds.
Hint 2 · The first line
Write $s_{B} - s_{A}$ as a quadratic in $t$ . Level means it is zero. Check each root is a time when both formulae hold: $B$ stops at $t = 34$ .
Hint 3 · The full method
The lead $s_{B} - s_{A}$ is greatest when its derivative is zero, which is when the two cars have the same speed. Or complete the square.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can derive the constant acceleration formulae from v = u + at and s = ½(u + v)t.
I can solve a quadratic in t and say what each root means, rejecting a root that cannot happen.
I can tell displacement from distance travelled when a particle changes direction.
I can set up equations for two particles that meet, or for motion in two stages.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers