Section 5

Assignment 5

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 5.

Links to the assessment requirements

The assignment uses exam-type questions and will test your understanding of the concepts you have explored in the topics listed above. These address the following in the Statistics and Mechanics part of the specification:

6.1, 7.1

7.2, 7.3, 8.3

6.1, 8.1, 8.28.3, 8.4

10.5 (pure core), 7.1, 8.2

7.4

How your tutor will mark your work

Your tutor will:

? check that you have answered each question ? check your workings as well as your answer ? give you feedback ? make suggestions about how you may be able to improve your work.

Are you ready to do the assignment?

Before you do the assignment, work through the topics in Section 5, completing the practice questions and summary exercises. By doing this you will cover all the concepts and techniques you will need for the

assignment.

What to do

Answer as many questions as you can.

It is always important to show your working in your responses.

Unless otherwise indicated, whenever a numerical value of g is required, take g = 9.8 ms–2 and give your answer to either 2 significant figures or 3 significant figures.

Guide time

The guide time for this assignment is 1 hour 30 minutes.

1 A ball is thrown from a point O so that its subsequent height above O, h m, is given by

h = (18x-3 )x2

where x m is its horizontal distance from O

(a) Find its horizontal distance from O when it is again at the same height as O (2)

(b) When x =2, calculate

(i) its height above O (2)

(ii) its distance from O correct to 1 decimal place. (2)

2 The velocity of a boat, v ms–1, is given by 4i – 3j, relative to a point O.

Find the time taken in minutes to reach a point 1.5?km away in the

direction of travel. (5)

3

The figure shows a lorry moving in a straight line with constant acceleration 0.2 ms–2.

The mass of the lorry is 5000?kg, it is driven by a constant force of

1200 N and the total resistance to motion is R N

Find the value of R (4)

GCE A level (AS/Part 1) Mathematics ? Section 5 ? Assignment 5

4

The figure above shows two particles A and B, connected by a light inextensible string which passes over a smooth pulley.

The mass of A is 3 kg and the mass of B is 4 kg.

The system is released from rest, B hits the floor after 3 seconds.

(a) Find the acceleration of the system when the particles are

released. (4)

(b) Find the original height of B from the floor. (3)

(c) Explain the significance of the string being light and inextensible in terms of modelling assumptions. (2)

5 A train starts from rest and accelerates uniformly at 0.2 ms–2 to reach its maximum speed in 2 minutes.

This speed is then maintained for the next 15 minutes.

The train is then brought to rest with uniform retardation 0.5 ms–2. (a) Sketch the velocity-time graph of the motion of the train. (2) Calculate:

(b) the maximum speed (2)

(c) the total distance travelled, in km to 3 significant figures. (4)

6 A helicopter flies from its base O to a boat B which has position vector 80i – 60j relative to O, where the units are kilometres.

It then flies directly in a straight line to a hospital H which has position vector 20i + 20j relative to O. uuur

(a) Find the vector BH . (2)

(b) Find the bearing on which it flies from B to arrive at H, taking the direction of j to be North, correct to the nearest

degree. (3)

(c) Given that it flies at a constant 250 kmh–1 and that it leaves B at

1240, find the time at which it reaches H. (4)

7 A particle P starts from a point O and moves in a straight line so that its velocity, v ms–1, is given by v = 8 + 2t – t2, where t is the time in seconds after leaving O.

Find:

(a) the values of t at the instants when the magnitude of the

acceleration is 1 ms–2 (4)

(b) the distance of P from O when t = 5 (3)

(c) the distance of P from O when P comes to instantaneous rest (3)

(d) the total distance travelled by P in the first 5 seconds. (2)

8 A smooth stone is thrown vertically from a point P with speed u ms–1 and then moves freely under gravity, returning to the point P after 6 seconds.

(a) Show that the value of u is 29.4 ms–1 to 3 s.f. (3)

(b) Find the greatest height reached by the stone. (3)

(c) Find the time for which the particle is above a height of 10 m

above P. (4)

9 Forces F1, F2 and F3, where F1 = (–2i + j) N, F2 = (4i + 2j) N and F3 = (ai + bj), act on a particle P.

(a) Given that the forces are in equilibrium, find the value of a and of b (3)

Given instead that the particle moves in the direction i + j

(b) show that a = b + 1 (3)

In this case, given further that the particle has mass 5?kg and the acceleration of P is 2 ms–2

(c) find the value of a and of b, given that they are both positive.

Your answers should be given in the form s + tv2, where s and t

are integers. (6)

(Total marks for Assignment 5 = 73)

Submit your assignment

When you have completed your assignment, submit it to your tutor for marking. You may need to scan your work, graphs and diagrams so that they are in a digital format, and save your document as a pdf file. Your

GCE A level (AS/Part 1) Mathematics ? Section 5 ? Assignment 5

tutor will send you helpful feedback and advice to help you progress through the course.

Assignment 5

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 5.

Links to the assessment requirements

The assignment uses exam-type questions and will test your understanding of the concepts you have explored in the topics listed above. These address the following in the Statistics and Mechanics part of the specification:

6.1, 7.1

7.2, 7.3, 8.3

6.1, 8.1, 8.28.3, 8.4

10.5 (pure core), 7.1, 8.2

7.4

How your tutor will mark your work

Your tutor will:

? check that you have answered each question ? check your workings as well as your answer ? give you feedback ? make suggestions about how you may be able to improve your work.

Are you ready to do the assignment?

Before you do the assignment, work through the topics in Section 5, completing the practice questions and summary exercises. By doing this you will cover all the concepts and techniques you will need for the

assignment.

What to do

Answer as many questions as you can.

It is always important to show your working in your responses.

Unless otherwise indicated, whenever a numerical value of g is required, take g = 9.8 ms–2 and give your answer to either 2 significant figures or 3 significant figures.

Guide time

The guide time for this assignment is 1 hour 30 minutes.

1 A ball is thrown from a point O so that its subsequent height above O, h m, is given by

h = (18x-3 )x2

where x m is its horizontal distance from O

(a) Find its horizontal distance from O when it is again at the same height as O (2)

(b) When x =2, calculate

(i) its height above O (2)

(ii) its distance from O correct to 1 decimal place. (2)

2 The velocity of a boat, v ms–1, is given by 4i – 3j, relative to a point O.

Find the time taken in minutes to reach a point 1.5?km away in the

direction of travel. (5)

3

The figure shows a lorry moving in a straight line with constant acceleration 0.2 ms–2.

The mass of the lorry is 5000?kg, it is driven by a constant force of

1200 N and the total resistance to motion is R N

Find the value of R (4)

GCE A level (AS/Part 1) Mathematics ? Section 5 ? Assignment 5

4

The figure above shows two particles A and B, connected by a light inextensible string which passes over a smooth pulley.

The mass of A is 3 kg and the mass of B is 4 kg.

The system is released from rest, B hits the floor after 3 seconds.

(a) Find the acceleration of the system when the particles are

released. (4)

(b) Find the original height of B from the floor. (3)

(c) Explain the significance of the string being light and inextensible in terms of modelling assumptions. (2)

5 A train starts from rest and accelerates uniformly at 0.2 ms–2 to reach its maximum speed in 2 minutes.

This speed is then maintained for the next 15 minutes.

The train is then brought to rest with uniform retardation 0.5 ms–2. (a) Sketch the velocity-time graph of the motion of the train. (2) Calculate:

(b) the maximum speed (2)

(c) the total distance travelled, in km to 3 significant figures. (4)

6 A helicopter flies from its base O to a boat B which has position vector 80i – 60j relative to O, where the units are kilometres.

It then flies directly in a straight line to a hospital H which has position vector 20i + 20j relative to O. uuur

(a) Find the vector BH . (2)

(b) Find the bearing on which it flies from B to arrive at H, taking the direction of j to be North, correct to the nearest

degree. (3)

(c) Given that it flies at a constant 250 kmh–1 and that it leaves B at

1240, find the time at which it reaches H. (4)

7 A particle P starts from a point O and moves in a straight line so that its velocity, v ms–1, is given by v = 8 + 2t – t2, where t is the time in seconds after leaving O.

Find:

(a) the values of t at the instants when the magnitude of the

acceleration is 1 ms–2 (4)

(b) the distance of P from O when t = 5 (3)

(c) the distance of P from O when P comes to instantaneous rest (3)

(d) the total distance travelled by P in the first 5 seconds. (2)

8 A smooth stone is thrown vertically from a point P with speed u ms–1 and then moves freely under gravity, returning to the point P after 6 seconds.

(a) Show that the value of u is 29.4 ms–1 to 3 s.f. (3)

(b) Find the greatest height reached by the stone. (3)

(c) Find the time for which the particle is above a height of 10 m

above P. (4)

9 Forces F1, F2 and F3, where F1 = (–2i + j) N, F2 = (4i + 2j) N and F3 = (ai + bj), act on a particle P.

(a) Given that the forces are in equilibrium, find the value of a and of b (3)

Given instead that the particle moves in the direction i + j

(b) show that a = b + 1 (3)

In this case, given further that the particle has mass 5?kg and the acceleration of P is 2 ms–2

(c) find the value of a and of b, given that they are both positive.

Your answers should be given in the form s + tv2, where s and t

are integers. (6)

(Total marks for Assignment 5 = 73)

Submit your assignment

When you have completed your assignment, submit it to your tutor for marking. You may need to scan your work, graphs and diagrams so that they are in a digital format, and save your document as a pdf file. Your

GCE A level (AS/Part 1) Mathematics ? Section 5 ? Assignment 5

tutor will send you helpful feedback and advice to help you progress through the course.

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