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5th Year Applied Maths Higher Level Kieran Mills
Uniform Accelerated Motion
No part of this publication may be copied, reproduced or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior written permission from The Dublin School of Grinds.
Ref: 5/appmaths/h/km/UAM
Oral Preparation CoursesSeparate to the Easter Revision Courses, The Dublin School of Grinds is also running Oral Preparation Courses. With the Oral marking component of the Leaving Certificate worth up to 40%, it is of paramount importance that students are fully prepared for these examinations. These courses will show students how to lead the Examiner towards topics that the student is prepared in. This will provide students with the confidence they need to perform at their peak.
ORAL PREPARATION COURSE FEES:
PRICE TOTAL SAVINGS
1st Oral Course €140 €140 -
2nd Oral Course €100 €240 €40
Looking to maximise your CAO points?Easter is well known as a time for students to vastly improve on the points that they received in their mock exams. To help students take advantage of this valuable time, The Dublin School of Grinds is running intensive exam-focused Easter Revision Courses. Each course runs for five days (90 minutes per day).
The focus of these courses is to maximise students’ CAO points. Special offer: Buy 1st course and get 2nd course free. To avail of this offer, early booking is required as courses were fully booked last year.
What do students get at these courses?
9 90 minutes of intensive tuition per day for five days, with Ireland’s leading teachers.
9 Comprehensive study notes.
9 A focus on simple shortcuts to raise students’ grades and exploit the critically important marking scheme.
To book, call us on 01-442 4442 or book online at www.dublinschoolofgrinds.ie
NOTE: These courses are built on the fact that there are certain predicable trends that appear and reoccur over and over again in the State Examinations.
FREE DAILY BUS SERVICE For full information on our Easter bus service, see 3 pages ahead.
NOTE: Any bookings for Junior Cert courses will also receive a weekly grind in one subject for the rest of the academic year, free of charge. This offer applies to 3rd and 2nd year students ONLY.
Timetable An extensive range of course options are available over a two-week period to cater for students’ timetable needs. Courses are held over the following weeks:
» Monday 21st March – Friday 25th March 2016» Monday 28th March – Friday 1st April 2016
All Easter Revision Courses take place in The Talbot Hotel, Stillorgan (formerly known as The Stillorgan Park Hotel).
BOOK EARLY TO AVAIL OF THE SPECIAL OFFER
BUY 1ST COURSE GET 2ND COURSE
F R E E ! Due to large course content, these subjects have been
divided into two courses. For a full list of topics covered in these courses, please see 3 pages ahead.
*
6th Year Easter Revision CoursesSUBJECT LEVEL DATES TIME
Accounting H Monday 21st March – Friday 25th March 8:00am - 9:30am
Agricultural Science H Monday 28th March – Friday 1st April 2:00pm - 3:30pm
Applied Maths H Monday 28th March – Friday 1st April 8:00am - 9:30am
Art History H Monday 28th March – Friday 1 April 8:00am - 9:30am
Biology Course A* H Monday 21st March – Friday 25th March 8:00am - 9:30am
Biology Course A* H Monday 21st March – Friday 25th March 12:00pm - 1:30pm
Biology Course A* H Monday 28th March – Friday 1st April 10:00am - 11:30am
Biology Course B* H Monday 21st March – Friday 25th March 10:00am - 11:30am
Biology Course B* H Monday 21st March – Friday 25th March 2:00pm - 3:30pm
Biology Course B* H Monday 28th March – Friday 1st April 8:00am - 9:30am
Business H Monday 21st March – Friday 25th March 12:00pm - 1:30pm
Business H Monday 28th March – Friday 1st April 8:00am - 9:30am
Chemistry Course A* H Monday 28th March – Friday 1st April 12:00pm - 1:30pm
Chemistry Course B* H Monday 28th March – Friday 1st April 2:00pm - 3:30pm
Classical Studies H Monday 21st March – Friday 25th March 8:00am - 9:30am
Economics H Monday 21st March – Friday 25th March 8:00am - 9:30am
Economics H Monday 28th March – Friday 1st April 10:00am - 11:30am
English Paper 1* H Monday 21st March – Friday 25th March 12:00pm - 1:30pm
English Paper 2* H Monday 21st March – Friday 25th March 10:00am - 11:30am
English Paper 2* H Monday 21st March – Friday 25th March 2:00pm - 3:30pm
English Paper 2* H Monday 28th March – Friday 1st April 10:00am - 11:30am
English Paper 2* H Monday 28th March – Friday 1st April 12:00pm - 1:30pm
French H Monday 21st March – Friday 25th March 10:00am - 11:30am
French H Monday 28th March – Friday 1st April 8:00am - 9:30am
Geography H Monday 28th March – Friday 1st April 8:00am - 9:30am
Geography H Monday 28th March – Friday 1st April 10:00am - 11:30am
German H Monday 21st March – Friday 25th March 10:00am - 11:30am
History (Europe)* H Monday 21st March – Friday 25th March 2:00pm - 3:30pm
History (Ireland)* H Monday 21st March – Friday 25th March 12:00pm - 1:30pm
Home Economics H Monday 21st March – Friday 25th March 10:00am - 11:30am
Irish H Monday 21st March – Friday 25th March 10:00am - 11:30am
Irish H Monday 28th March – Friday 1st April 12:00pm - 1:30pm
Maths Paper 1* H Monday 21st March – Friday 25th March 8:00am - 9:30am
Maths Paper 1* H Monday 21st March – Friday 25th March 12:00pm - 1:30pm
Maths Paper 1* H Monday 28th March – Friday 1st April 10:00am - 11:30am
Maths Paper 1* H Monday 28th March – Friday 1st April 2:00pm - 3:30pm
Maths Paper 2* H Monday 21st March – Friday 25th March 10:00am - 11:30am
Maths Paper 2* H Monday 21st March – Friday 25th March 2:00pm - 3:30pm
Maths Paper 2* H Monday 28th March – Friday 1st April 12:00pm - 1:30pm
Maths Paper 2* H Monday 28th March – Friday 1st April 4:00pm - 5:30pm
Maths O Monday 21st March – Friday 25th March 8:00am - 9:30am
Maths O Monday 28th March – Friday 1st April 12:00pm - 1:30pm
Physics H Monday 28th March – Friday 1st April 10:00am - 11:30am
Spanish H Monday 21st March – Friday 25th March 2:00pm - 3:30pm
Spanish H Monday 28th March – Friday 1st April 10:00am - 11:30am
6th Year Oral Preparation CoursesSUBJECT LEVEL DATES TIME
French H Sunday 20th March 10:00am - 2:00pm
German H Saturday 26th March 10:00am - 2:00pm
Irish H Saturday 26th March 10:00am - 2:00pm
Spanish H Saturday 19th March 1:00pm - 5:00pm
5th Year Easter Revision CoursesSUBJECT LEVEL DATES TIME
Maths H Monday 28th March – Friday 1st April 8:00am - 9:30am
English H Monday 28th March – Friday 1st April 4:00pm - 5:30pm
Note: 5th year students are welcome to attend any 6th year course as part of our buy 1 get 1 free offer.
3rd Year Easter Revision CoursesSUBJECT LEVEL DATES TIME
Business Studies H Monday 28th March – Friday 1st April 8:00am - 9:30am
English H Monday 21st March – Friday 25th March 8:00am - 9:30am
English H Monday 28th March – Friday 1st April 2:00pm - 3:30pm
French H Monday 28th March – Friday 1st April 12:00pm - 1:30pm
Geography H Monday 28th March – Friday 1st April 12:00pm - 1:30pm
German H Monday 21st March – Friday 25th March 8:00am - 9:30am
History H Monday 21st March – Friday 25th March 4:00pm - 5:30pm
Irish H Monday 28th March – Friday 1st April 2:00pm - 3:30pm
Maths H Monday 21st March – Friday 25th March 10:00am - 11:30am
Maths H Monday 21st March – Friday 25th March 12:00pm - 1:30pm
Maths H Monday 28th March – Friday 1st April 10:00am - 11:30am
Maths O Monday 28th March – Friday 1st April 12:00pm - 1:30pm
Science H Monday 28th March – Friday 1st April 2:00pm - 3:30pm
Science H Monday 21st March – Friday 25th March 2:00pm - 3:30pm
Spanish H Monday 21st March – Friday 25th March 12:00pm - 1:30pm
2nd Year Easter Revision CoursesSUBJECT LEVEL DATES TIME
Maths H Monday 21st March – Friday 25th March 2:00pm - 3:30pm
NOTE: Any bookings for Junior Cert courses will also receive a weekly grind in one subject for the rest of the academic year, free of charge. This offer applies to 3rd and 2nd year students ONLY.
Contents: Uniform ACCelerAted motion
seCtion 1 Using the FormUlae ...................................................................2Exercise 1 ..................................................................................6Exercise 2 ..................................................................................8
seCtion 2 sUccessive times and distances ................................................10Exercise 3 ..................................................................................14
1. A train starts from rest and accelerates uniformly at 2.5 m s–2 until it reaches a speed of25 m s–1. Find the distance moved and the time taken for this motion.
2. A car can accelerate from rest to 90 km h–1 in 7.5 seconds. Find its acceleration.
3. In travelling 65 cm along the barrel of a rifle a bullet accelerates from rest to 230 m s–1.Find the accceleration and the time the bullet is in the barrel.
4. A car travelling at 24 m s–1 requires a minimum braking distance of 36 m. What is itsdeceleration? How long does it take to stop?
5. A car starts from rest with acceleration 4 m s–2. How far does it go in (i) 2 s, (ii) 3 s,(iii) the third second.
6. A body moves in a straight line and increases its velocity from 3 m s–1 to 15 m s–1
uniformly in 6 s. Find the acceleration and the distance travelled.
7. A particle starts with a velocity of 3 m s–1 and accelerates uniformly at 1.5 m s–2.How far does it go in (i) 1 s, (ii) 5 s, (iii) the fourth second.
8. A body is projected from the origin with a velocity of 8 m s–1 and acceleration –2 m s–2.Find(i) the velocity when t = 3 s,(ii) when it comes to instantaneous rest.
9. A particle moves along a straight line between two points P and Q with constantacceleration 0.8 m s–2. Its velocity at Q is 1.2 m s–1 greater than the velocity at P.If the distance PQ is 48 m, find the velocity at P.How long after passing P does it take the velocity to reach 48 m s–1.
10. A car is moving with speed u m s–1. The brakes of the car can produce a constantdeceleration of 5 m s–2. It is known that when the driver decides to stop, a period of 2
5 selapses before the brakes are applied. As the car passes a point O, the driver decides tostop.Find in terms of u the minimum distance of the car from O when the car comes to rest.The driver is approaching traffic lights and is 102 m away when the light changes fromgreen to amber. The lights remain amber for 3 s before changing to red.Show(a) when u < 30 the driver can stop before reaching the lights,(b) when u > 34 the driver can pass the light before it turns red.
1. In two successive seconds a uniformly accelerating body travels 4 m and 8 m.Find its acceleration.
2. A uniformly accelerating body travels 5 m and 11 m repectively in its first two seconds.How far does it travel in the fourth second?
3. A uniformly decelerating body covers successive 100 m distances in 5 s and 10 s.Find its initial speed, the deceleration and the further time for the body to come to rest.
4. A particle starts from rest and moves in a straight line with uniform acceleration.It passes three points A, B and C where |AB| =105 m and |BC| = 63 m. If it takes 6 s to travelfrom A to B and 2 s from B to C find(i) its acceleration, (ii) the distance of A from the starting position.
5. A sprinter runs a race with constant acceleration throughout. During the race he passes fourposts A, B, C, D such that |AB| = |BC| = |CD| = 36 m. If the sprinter takes 3 s to run from A toB and 2 s to run from B to C, how long does it takes to run from C to D?
6. A particle moving in a straight line with uniform acceleration describes 23 m in the fifthsecond of its motion and 31 m in the seventh second. Calculate its initial velocity.
7. A body travels in a straight line with uniform acceleration. The particle passes three points A,B and C at t = 0, t = 3 s and t = 6 s. If |BC| = 90 m and the speed of the particle at B is21 m s–1, find the acceleration of the body and its speed at A.
8. A, B, C are three points which lie in that order on a straight road with |AB| = 45 m and|BC| = 32 m. A car travels along the road in the direction ABC with constant acceleration f.The car passes A with speed u and passes B five seconds later and passes C two seconds afterthat. Find u and f.
9. A car is moving along at a steady 20 m s–1 when the driver suddenly sees a tree across theroad 56 m ahead. He immediately applies the brakes giving the car a constant deceleration of4 m s–2. How far in front of the tree does the car come to rest? If the driver had not reactedimmediately and the brakes were applied one second later with what speed would the carhave hit the tree?
10. A, B, C are three points on a straight line in that order. A body is projected from B towards Awith a speed of 5 m s–1. The body experiences an acceleration of 2 m s–2 towards C.If |BC| = 24 m, find the time to reach C, and the distance travelled by the body from theinstant of projection until it reaches C.
11. A bus 12.5 m long travels with constant acceleration. The front of the bus passes a point Pwith speed u and the rear passes P with speed v. Find in terms of u and v(i) the time taken for the bus to pass P,(ii) what fraction of the bus passes P in half this time.
12. A body moving in a straight line with constant acceleration passes in succession throughpoints A, B, C and D where |AB| = x, |BC| = y and |CD| = z where the distances x, y and z arecovered in equal intervals of time. Show 2y = x + z.
13. A uniformly decelerating train of length 40 m enters a station of length 80 m. The frontengine leaves the station 5 s later and the rear of the train leaves the station after a further5 s. Find the deceleration of the train.
14. A uniformly accelerating body starts with a speed of u, in successive times of t travels
distances s and 2s. Prove that its acceleration is 4 2us
.
15. A body starts moving in a straight line with velocity u and acceleration a. If when thevelocity has increased to 5u the acceleration is reversed in direction its magnitude beingunaltered prove that when the particle returns to its starting point its velocity will be –7u.
ExErcisE 4. catch up1. Two bodies start together at the same time at the same place and move along the same
straight line. If one moves with a constant speed of 8 m s–1 while the other starts from restand moves at a constant acceleration of 2 m s–2. How long will it take before they aretogether?
2. A car A passes a point P on a straight road at a constant speed of 10 m s–1. At the same timeanother car B starts from rest at P with uniform acceleration 2.5 m s–2.(i) When and how far from P will B overtake A.(ii) If B ceases to accelerate on overtaking, what time elapses between the two cars passing a
point Q which is 3 km from P.
3. A boy runs at 4 m s–1 away from a cyclist who starts at rest and accelerates at2 m s–2. If the boy has an initial lead of 5 m, how long does the cyclist take to catch him?
4. Two bodies A and B travel in the same direction along the same line. Body A starts withvelocity 3 m s–1 and acceleration 2 m s–2. The other body starts from the same place withvelocity 1 m s–1 and acceleration 3 m s–2. Find when and where they are together again.
5. Two bodies move along parallel tracks in the same direction. Body A starts with velocity2 m s–1 and acceleration 6 m s–2. Body B starts from the same place and the same time withvelocity 5 m s–1 and acceleration 2 m s–2. Find when and where they are together again. Findtheir velocities when they are together for the second time.
6. Two bodies move in the same direction along parallel paths. A starts from point O withvelocity 2 m s–1 and acceleration 4 m s–2. B starts 6 m ahead of A with velocity 3 m s–1 andacceleration 2 m s–2. Find when and where they are together and their velocities at thisinstant.
7. Two bodies move in the same direction along parallel paths. A starts from a point O withvelocity 8 m s–1 and acceleration 2 m s–2 and B starts 8 m ahead of A and moves off withvelocity 2 m s–1 at acceleration 4 m s–2. Find when they will be together and their distancesfrom O at these times.
8. Two bodies A and B travel in the same direction along the same line from the same point P atthe same time. A starts with velocity 5 m s–1 and acceleration 3 m s–2. B starts with velocity2 m s–1 and acceleration 4 m s–2. They are together again at point Q. Find the time at whichthey are together and the distance | PQ |. Find their maximum distance apart between P and Q.
9. Two bodies move in the same direction along parallel paths. They start at the same point P atthe same time. A starts from P with velocity 3 m s–1 and acceleration 2 m s–2. B starts withvelocity 1 m s–1 and acceleration 3 m s–2. They are together again at point Q. Find the time atwhich they are together and the distance | PQ |.Find their maximum distance apart between Pand Q.
10. Two bodies A and B move along parallel straight lines in the same direction from the samepoint P. A starts with velocity 4 m s–1 and acceleration 2 m s–2. B starts 1 second after A withvelocity 2 m s–1 and acceleration 4 m s–2. Find when and where they will be together.
11. Two bodies A and B move along parallel straight lines in the same direction from the samepoint P. A starts from point P with velocity 5 m s–1 and acceleration 4 m s–2. B starts 1second before A with velocity 6 m s–1 and acceleration 3 m s–2 from a point a distance of2.5 m to the right of P. Find when and where they are together.
12. Find when and where they are together.Find their maximum separation between thetwo times when they are together.
13. IfAstarts2secondsbeforeBfindwhenandwhere they are together. Find their maximumseparation between the two positions.
14. A car A starts from a point P with initial velocity 8 m s–1 and then travels with uniformacceleration 4 m s–2. Two seconds later a second car B starts from P with an initial velocityof 30 m s–2 and then moves with a uniform acceleration of 3 m s–2. Show that after passing A,B will never be ahead by more than 74 m.
15. Bodies A and B start together and move along the same straight line. A starts with a speed of10 m s–1 and moves with a constant deceleration, while B starts at 5 m s–1 and accelerates at4 m s–2. Find the deceleration of A if they meet when the velocity of B is twice that of A.
16. The driver of a car travelling at 20 m s–1 sees a second car 120 m in front travelling in thesame direction at a uniform speed of 8 m s–1.(a) What is the least uniform retardation that must be applied to the faster car to avoid
collision?(b) If the actual retardation is 1 m s–2find
(i) the time interval in seconds for the faster car to reach a point 66 m behind the slowercar,
1. A car is travelling at 72 km h–1 when the brakes are applied producing a retardation of4 m s–2. How long does it take to stop?
2. An electric train starts from a station and reaches a speed of 14 m s–1 in 25 s with uniformacceleration.Sketchthevelocity-timegraph,andfindhowfarithasgonebythetimeitreaches this speed.
3. An aircraft can take off when it reaches a speed of 180 km h–1. If it attains this speed in 30 swith uniform acceleration what distance does it require for taking off?
4. An express train is travelling at 144 km h–1 when its brakes are applied. If these produce aretardation of 2 m s–2 how long will it take to stop and what distance will it cover in doingso?
5. A train starts from rest and attains a speed of 50 km h–1 in 4 minutes with uniformacceleration. It runs at that speed for 5 minutes and then slows down uniformly to rest in2minutes.Drawthevelocity-timegraphandfindthetotaldistancetravelled.
6. Find from the velocity-time graph shown(i) theaccelerationduringthefirst4s,(ii) the retardation during the last 2 s,(iii) the total distance travelled.
7. A cyclist rides along a straight road from A to B. He starts from rest at A and acceleratesuniformly to reach a speed of 10 m s–1 in 8 s. He maintains this speed for 30 s and thenuniformly decelerates to rest at B. If the total time is 48 s, draw a velocity-time curve andfromitfind(i) the acceleration,(ii) the deceleration,(iii) the total distance travelled.
8. A car travels from A to B. It starts from rest at A and accelerates at 1.5 m s–2 until it reaches aspeed of 30 m s–1. It then travels at this speed for 2 km and then decelerates at 2 m s–2 tocome to rest at B. Find(i) the total time for the journey,(ii) the distance from A to B,(iii) the average speed for the journey.
9. A and B are two points on a straight road. A car travelling along the road passes A whent = 0 and maintains a constant speed until t=20s,andinthistimecoversfour-fifthsofthedistance from A to B. The car then decelerates uniformly to rest at B. Draw a velocity-timecurveandfindthetimefromAtoB.
10. A tram travels along a straight track and starts from rest. It accelerates uniformly for 20 sand during this time it travels 160 m. It maintains a constant speed for a further 50 s anddecelerates to rest in 8 s.Calculate(i) the acceleration,(ii) the deceleration,(iii) the total time,(iv) the total distance.
11. A train starts from rest and travels 8 km in 12 minutes ending at rest. The acceleration is halfthe retardation, both are uniform, and there is a period when the train runs at its maximumspeed of 50 km h–1. Find the time taken to reach full speed.
12. A 100 m sprinter starts with a speed of 6 m s-1 and accelerates uniformly to 10 m s–1 andfinishestheraceatthisspeed.Ifhistotaltimeis10.4s,findhisuniformaccelerationandafter what distance he is going at full speed.
13. Acartakes2minutestotravelbetweentwosetsoftrafficlights2145mapart.Ithasuniformacceleration for 30 s, then uniform velocity, and then uniform retardation for the last 15 s.Find the maximum velocity and the acceleration.
14. A train travels 15 km between two stations at an average speed of 50 km h–1. Its accelerationis half the retardation and both are uniform. If the maximum speed is 72 km h–1findtheacceleration in m s–2. Sketch the velocity-time curve.
15. A car accelerates at 2 m s–2 in bottom gear, 1.5 m s–2 in second gear and 1 m s–2 in top gear.Each gear change takes 1.5 s during which time the car travels at constant speed. If amotorist changes gear when his speeds are 3 m s–1 and 9 m s–1findhowlonghewilltaketoreach 15 m s–1 from rest.
16. A train moving in a straight line starts from A with uniform acceleration of 0.1 m s–2. After ithas attained full speed it moves uniformly for 10 minutes. It is brought to rest at B by thebrakes, which apply a constant retardation of 0.8 m s–2 for 20 s. Draw a rough velocity-timegraphandfromitfindthetimeofthejourneyandthedistancefromA to B.
17. A train has a maximum speed of 72 km h–1 which it can achieve at an acceleration of0.25 m s–2. With its brakes fully applied the train has a deceleration of 0.5 m s–2. What is theshortest time that the train can travel between stations 8 km apart if it stops at both stations?
18. A particle with speed 150 m s–1 begins to decelerate uniformly at a certain instant whileanother particle starts from rest 8 s later and accelerates uniformly. When the second particlehas travelled 135 m, both particles have a speed of 30 m s–1.(i) Show the motion of both on the same speed-time curve.(ii) Howmanysecondsafterthecommencementofdecelerationdoesthefirstparticlecome
19. A body starts from rest at P travelling in a straight line and then comes to rest at Q which is696 m from P.Thetimetakenis66s.Forthefirst10sithasuniformaccelerationa. It thentravelsatconstantspeedandisfinallybroughttorestbyauniformdecelerationb acting for6 s. Find a and b.If the journey from rest at P to rest at Q had been travelled with no interval of constant speedbut at acceleration of a for a time t1 immediately followed by deceleration b for a time t2,show that the time for the journey is 8 29 s.
20. An athlete runs 100 m in 12 s. Starting from rest he accelerates uniformly to a speed of10 m s–1 and then continues at that speed. Calculate the acceleration.
21. A cyclist has a maximum acceleration of 2 m s–2, a maximum speed of 15 m s–1 and amaximum deceleration of 4 m s–2. If he travels from rest to rest in the shortest possible timeshow that he covers a distance of 84 3
1. A vase falls from a shelf 140 cm above the floor. Find the speed with which it strikes thefloor.
2. A stone is dropped from a point 49 m above the ground. Find the time for it to reach theground.
3. A stone is thrown down at 5 m s–1. If its speed on hitting the ground is 19 m s–1 from whatheight was it thrown. How long does it take?
4. A stone is dropped from the top of a tower and falls to the ground. If it strikes the ground at14 m s–1, how high is the tower?
5. A ball is thrown vertically downwards from the top of a tower with an initial speed of2 m s–1. If it hits the ground 3 s later find(i) the height of the tower,(ii) the speed with which it hits the ground.
6. A stone is thrown upwards with a speed of 21 m s–1. Find its height(i) 1 s after projection,(ii) 2 s after projection,(iii) 3 s after projection.
7. A ball is thrown up at 14 m s–1 from a point 2 m above the ground. Find(i) the speed when it returns to the level of projection,(ii) the speed on the ground.
8. A ball is thrown vertically up at 28 m s–1. Find(i) the maximum height,(ii) the time to reach the maximum height,(iii) the velocity of return,(iv) the total time for the journey.
9. A balloon is rising at a steady speed of 3 m s–1. How high is it above the ground after 10 s?At this instant a man releases a stone. What is the initial velocity of the stone?How long does it take to reach the ground? How high is the balloon above the ground whenthe stone strikes the ground?
10. A stone is thrown up at 49 m s–1 from the ground. Find the times at which the particle is78.4 m above the ground. Find the time interval for which the particle is above 78.4 m.
11. A ball is thrown up at 14 m s–1. Find the times at which the particle is 9.1 m above theground.
12. A ball is thrown up at 49 m s–1. How long does it take to reach its maximum height? If another ball was thrown up 1 s after the first one, how high is it above the ground when the firstball has reached its maximum height if it has the same initial velocity?
13. A jumper can jump 2 m on the Earth. What is his take-off speed? How high can he jump onthe moon? (Acceleration due to gravity of moon g = 1.6 m s–1 )
14. A particle is thrown vertically upwards under gravity with a speed of 16 m s–1. One secondlater another particle is fired upwards from the same point. Find the initial speed of thisparticle in order that the two particles will collide when the first particle has reached itshighest point.
15. An object falls vertically past a window 2 m high in 112 s. Find the height above the window
from which the object was dropped.
16. A stone is dropped from a balloon rising at 10 m s–1 and reaches the ground in8 s. How high was the balloon above the ground when the stone was dropped?
17. A body falls from the top of a tower and during the last second it falls 925 of the total
distance. Find the height of the tower.
18. A particle falls freely from rest from a point O passing three points A, B and C, the distances|AB| and |BC| being equal. If the particle takes 3 s to pass from A to B and 2 s from B to C,calculate |AB|.
19. A body falls freely from rest from a point O passing three points A, B and C, the distances|AB| and |BC| being equal. The time taken to go from A to B is 2 s and from B to C is 1 s.Find |AB|.
20. A particle falls freely under gravity from rest at a point P. After it has fallen for 1 s anotherparticle is projected vertically downwards from P with speed 14.7 m s–1. Find the time anddistance from P at which they collide.
20131. (a) A ball is thrown vertically upwards with a speed of 44·1 m s−1.
Calculate the time interval between the instants that the ball is 39·2 m above the pointof projection.
(b) A lift ascends from rest with constant acceleration f until it reaches a speed v. Itcontinues at this speed for t1 seconds and then decelerates uniformly to rest withdeceleration f.The total distance ascended is d, and the total time taken is t seconds.
(i) Draw a speed-time graph for the motion of the lift.
(ii) Show that v f t t= -12 1( ).
(iii) Show that t t df1
2 4= − .
LeAving cert Questions20141. (a) Two cars, P and Q, travel with the same constant velocity 15 m s–1 along a straight level
road. The front of car P is 24 m behind the rear of car Q. At a given instant both cars decelerate, P at 4 m s–2 and Q at 5 m s–2.
(i) Find, in terms of t, the distance between the cars t seconds later.
(ii) Find the distance between the cars when they are at rest.
20121. (a) A particle falls from rest from a point P. When it has fallen a distance 19·6 m a second
particle is projected vertically downwards from P with initial velocity 39·2 m s–1.The particles collide at a distance d from P.Find the value of d.
(b) A car, starts from rest at A, and accelerates uniformly at 1 m s–2 along a straight level roadtowards B, where AB = 1914 m. When the car reaches its maximum speed of 32 m s–1, itcontinues at this speed for the rest of the journey.At the same time as the car starts from A a bus passes B travelling towards A with aconstant speed of 36 m s–1. Twelve seconds later the bus starts to decelerate uniformly at0·75 m s–2.
(i) The car and the bus meet after t seconds. Find the value of t.
(ii) Find the distance between the car and the bus after 48 seconds.
20111. (a) A particle is released from rest at A and falls vertically passing
two points B and C.
It reaches B after t seconds and takes 27t
seconds to fall from B to C, a distance of 2.45 m.
Find the value of t.
(b) A car accelerates uniformly from rest to a speed v in t1 seconds.It continues at this constant speed for t seconds and then deceleratesuniformly to rest in t2 seconds.
The average speed for the journey is 34v .
(i) Draw a speed-time graph for the motion of the car.
(ii) Find t1 + t2 in terms of t.
(iii) If a speed limit of 23v were to be applied, find in terms of t the least time the
journey would have taken, assuming the same acceleration and deceleration as inpart (ii).
20101. (a) A car is travelling at a uniform speed of 14 ms–1 when the driver notices a traffic light
turning red 98 m ahead.
Find the minimum constant deceleration required to stop the car at the traffic light,(i) if the driver immediately applies the brake(ii) if the driver hesitates for 1 second before applying the brake.
(b) A particle passes P with speed 20 ms–1 and moves in a straight line to Q with uniformacceleration.In the first second of its motion after passing P it travels 25 m.In the last 3 seconds of its motion before reaching Q it travels 13
20 of |PQ|.
Find the distance from P to Q.
20091. (a) A particle is projected vertically upwards from
the point P. At the same instant a second particleis let fall vertically from Q.The particles meet at R after 2 seconds.
The particles have equal speeds when they meetat R.
Prove that |PR| = 3|RQ|.
(b) A train accelerates uniformly from rest to a speed v m/s with uniformacceleration f m/s2.
It then decelerates uniformly to rest with uniform retardation 2f m/s2.The total distance travelled is d metres.(i) Draw a speed-time graph for the motion of the train.
(ii) If the average speed of the train for the whole journey is d3
20081. (a) A ball is thrown vertically upwards with an initial velocity of 39.2 m/s. Find
(i) the time taken to reach the maximum height
(ii) the distance travelled in 5 seconds.
(b) Two particles P and Q, each having constant acceleration, are moving in the samedirection along parallel lines. When P passes Q the speeds are 23 m/s and 5.5 m/s,respectively. Two minutes later Q passes P, and Q is then moving at 65.5 m/s. Find(i) the acceleration of P and the acceleration of Q
(ii) the speed of P when Q overtakes it
(iii) the distance P is ahead of Q when they are moving with equal speeds.
20071. (a) A particle is projected vertically downwards from the top of a
tower with speed u m/s. It takes the particle 4 seconds to reach the bottom of the tower.During the third second of its motion the particle travels 29.9 metres.
Find(i) the value of u
(ii) the height of the tower.
(b) A train accelerates uniformly from rest to a speed v m/s.It continues at this speed for a period of time and then decelerates uniformly to rest.In travelling a total distance d metres the train accelerates through a distancepd metres and decelerates through a distance qd metres, where p < 1 and q < 1.(i) Draw a speed-time graph for the motion of the train.
(ii) If the average speed of the train for the whole journey is vp q b+ +
20061. (a) A lift starts from rest. For the first part of its descent it travels with uniform
acceleration f. It then travels with uniform retardation 3f and comes to rest.The total distance travelled is d and the total time taken is t.(i) Draw a speed-time graph for the motion.
(ii) Find d in terms of f and t.
(b) Two trains P and Q, each of length 79.5 m, moving in opposite directions along parallellines, meet at O, when their speeds are 15 m/s and 10 m/s respectively.The acceleration of P is 0.3 m/s2 and the acceleration of Q is 0.2 m/s2.It takes the trains t seconds to pass each other.(i) Find the distance travelled by each train in t seconds.
(ii) Hence, or otherwise, calculate the value of t.
(iii) How long does it take for 25 of the length of train Q to pass the point O?
20051. (a) Car A and car B travel in the same direction along a horizontal straight road.
Each car is travelling at a uniform speed of 20 m/s.Car A is at a distance of d metres in front of car B.At a certain instant car A starts to brake with a constant retardation of 6 m/s2.0.5 s later car B starts to brake with a constant retardation of 3 m/s2 .Find(i) the distance travelled by car A before it comes to rest
(ii) the minimum value of d for car B not to collide with car A.
20041. (a) A ball is thrown vertically upwards with an initial velocity of 20 m/s.
One second later, another ball is thrown vertically upwards from the same point with an initial velocity of u m/s.The balls collide after a further 2 seconds.(i) Show that u = 17.75.
(ii) Find the distance travelled by each ball before the collision, giving your answerscorrect to the nearest metre.
20031. (a) The points P, Q and R all lie in a straight line.
A train passes point P with speed u m/s. The train is travelling with uniform retardation f m/s2. The train takes 10 seconds to travel from P to Q and 15 seconds to travel from Q to R, where | PQ | = | QR | = 125 metres.
(i) Show that f = 13 .
(ii) The train comes to rest s metres after passing R.Find s, giving your answer correct to the nearest metre.
(b) A man runs at constant speed to catch a bus.At the instant the man is 40 metres from the bus, it begins to accelerate uniformly fromrest away from him.The man just catches the bus 20 seconds later.(i) Find the constant speed of the man.
(ii) If the constant speed of the man had instead been 3 m/s, show that the closest he getsto the bus is 17.5 metres.
20021. (a) A stone is thrown vertically upwards under gravity with a speed of u m/s from a point
30 metres above the horizontal ground. The stone hits the ground 5 seconds later. (i) Find the value of u.
(ii) Find the speed with which the stone hits the ground.
(b) A particle, with initial speed u, moves in a straight line with constant acceleration.During the time interval from 0 to t, the particle travels a distance p.During the time interval from t to 2t, the particle travels a distance q.During the time interval from 2t to 3t, the particle travels a distance r.(i) Show that 2q = p + r.
(ii) Show that the particle travels a further distance 2r – q in the time interval from 3tto 4t.
20011. (a) Points P and Q lie in a straight line, where | PQ | = 1200 metres.
Starting from rest at P, a train accelerates at 1 m/s2 until it reaches the speed limit of 20 m/s. It continues at this speed of 20 m/s and then decelerates at 2 m/s2, coming to rest at Q. Find the time it takes the train to go from P to Q. Find the shortest time it takes the train to go from rest at P to rest at Q if there is no speed limit, assuming that the acceleration and deceleration remain unchanged at 1 m/s2
and 2 m/s2, respectively.
(b) A particle is projected vertically upwards with an initial velocity of u m/s andanother particle is projected vertically upwards from the same point and with the sameinitial velocity T seconds later.Show that the particles
(i) will meetT u
g2+
seconds from the instant of projection of the first particle
(ii) will meet at a height of4
8
2 2 2u g Tg−
metres.
20001. (a) A stone projected vertically upwards with an initial speed of u m/s rises 70 m in the first
t seconds and another 50 m in the next t seconds.Find the value of u.
(b) A car, starting from rest and travelling from P to Q on a straight level road, where| PQ | = 10 000 m, reaches its maximum speed 25 m/s by constant acceleration in the first500 m and continues at this maximum speed for the rest of the journey.A second car, starting from rest and travelling from Q to P, reaches the samemaximum speed by constant acceleration in the first 250 m and continues at thismaximum speed for the rest of the journey.(i) If the two cars start at the same time, after how many seconds do the two cars meet?
Find, also, the distance travelled by each car in that time.
(ii) If the start of one car is delayed so that they meet each other exactly halfwaybetween P and Q, find which car is delayed and by how many seconds.
19991 (b) A particle travels in a straight line with constant acceleration f for 2t seconds and covers
15 metres. The particle then travels a further 55 metres at constant speed in 5t seconds. Finally the particle is brought to rest by a constant retardation 3f.
(i) Draw a speed-time graph for the motion of the particle.
(ii) Find the initial velocity of the particle in terms of t.
(iii) Find the total distance travelled in metres, correct to two decimal places.
19981 (a) A train accelerates uniformly from rest to a speed v m/s. It continues at this constant
speed for a period of time and then decelerates uniformly to rest. If the average speed for the whole journey is 5
6 v, find what fraction of the whole distance is described at constant speed.
(b) Car A, moving with uniform acceleration 320 b m/s2 passes a point P with speed 9u m/s.
Three seconds later car B, moving with uniform acceleration 29 b m/s2 passes the same
point with speed 5u m/s. B overtakes A when their speeds are 6.5 m/s and 5.4 m/s respectively. Find(i) the value u and the value b,
(ii) the distance travelled from P until overtaking occurs.
19971 (a) A particle, moving in a straight line, accelerates uniformly from rest to a speed v m/s. It
continues at this constant speed for a time and then decelerates uniformly to rest, the magnitude of the deceleration being twice that of the acceleration. The distance travelled while accelerating is 6 m. The total distance travelled is 30 m and the total time taken is 6 s.
(i) Draw a speed-time graph and hence, or otherwise, find the value of v.
(ii) Calculate the distance travelled at v m/s.
(b) Two points P and Q are a distance d apart. A particle starts from P and move towards Qwith initial velocity 2u and uniform acceleration f. A second particle starts at the sametime from Q and moves towards P with initial velocity 3u and uniform deceleration f.Prove that
(i) the particles collide afterdu5
seconds,
(ii) if the particles collide before the second particle comes to instantaneous rest, thenfd u<15 2,
(iii) if fd u= 30 2 then the second particle has returned to Q before the collision.
19961 (a) A particle starts from rest and moves in a straight line with uniform acceleration. It
passes three points A, B and C where |AB| = 105 m and |BC| = 63 m. If it takes 6 seconds to travel from A to B and 2 seconds to travel from B to C find(i) its acceleration
(ii) the distance of A from the starting position.
(b) A lift starts from rest with constant acceleration 4 m/s2. It then travels with uniformspeed and finally comes to rest with constant retardation 4 m/s2. The total distancetravelled is d and the total time taken is t.(i) Draw a speed-time graph.
(ii) Show that the time for which it travelled with uniform speed is t d2 − .