Motion of Particles: Calculating Time, Speed, and Extension, Exams of Applied Mathematics

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Coimisiún na Scrúduithe Stáit

State Examinations Commission

Leaving Certificate 2011

Marking Scheme

Higher Level

APPLIED MATHEMATICS

1. (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 7

2 t seconds to

fall from B to C , a distance of 2.45 m.

Find the value of t.

t

t

t t

t gt g g

t h g

AC s ut ft

h gt

AB s ut ft

s 8

2

2 2

2 2

1 4

(^21) 2

1

2 2

1

2 2 1

2 2

1

2 2

1

A

B

C

1. (b) A car accelerates uniformly from rest to a speed v in t 1 seconds. It continues at this constant speed for t seconds and then decelerates uniformly to rest in t 2 seconds. The average speed for the journey is 4

3 v .

(i) Draw a speed-time graph for the motion of the car. (ii) Find t 1 + t 2 in terms of t. (iii) If a speed limit of 3

2 v were to be applied, find in terms of t the least

time the journey would have taken, assuming the same acceleration and deceleration as in part (ii).

(i)

t t t

t t t t t t

t t t

t t t

t t t

v tv tv tv

t t t

tv tv tv

1 2

1 2 1 2

1 2

2 2

1 2 1

1

1 2

2 2 1 2 1 1

1 2

2 2 1 2 1 1

(ii) averagespeed

 

t t t t t t

t t

t t t

t t t

t t t t

t t t t t t

tv tv tv tv tv tv

v t v t v tv tv tv t

12

31 12

23 3 2 4 5 6

12

23 5

3 5

2

3

2 4 6

4 6 5

1 2 4 5 6

1 2 4 5 6

2 6

1 2 4 5

1 2 2

1 21

1

(iii)

v

t 1 t t 2

32 v

t 4 t 5 t 6

2 (b) A woman can row a boat at 4 m s^1 in still water. She rows across a river 80 m wide. The river flows at a constant speed of 3 .5 ms^1 parallel to the straight banks. She wishes to land between B and C. The point B is directly across from the starting point A and the point C is 20 3 m downstream from B.

If  is the direction she takes, find the range of values of  if she lands between B and C.

os

c 

sin 0. 8029

sin

3. 5

sin

tan

B

B C



80 m 4

A



α

B C



β

A

3. (a) A particle is projected from a point P on horizontal ground.

The speed of projection is 35 m s^1 at an angle tan ^1 2 to the horizontal. The particle strikes a target whose position vector relative to P is x i j

Find (i) the value of x (ii) a second angle of projection so that the particle strikes the target.

35 sin. 4. 9 50

(i) 3 5cos.

2

2

2

x

x x

x x

t t

x t

t x

 

tan 3

tan 2 tan 3 0

tan 5 tan 6 0

50 tan 101 tan 50

7 cos

7 cos

35 sin.

35 sin. 4. 9 50

7 cos

(ii) 3 5cos. 50

2

2

2

2

t t

t

t

4. (a) A block of mass 2 2 kg rests on a rough plane inclined at 45º to the horizontal. It is connected by a light inextensible string which passes over a smooth, light, fixed pulley to a particle of mass 4 kg which hangs freely under gravity. The coefficient of friction between the block and the plane is 4

Find the acceleration of the 4 kg mass.

 

 

 

2

4 1

2. 15 ms

 ^ 

f

g f

f

g

g g g f

T g g f

T g R f

g T f

4g

μR

T

T

2 g

2 g

R

4 (b) A smooth pulley, of mass 2 kg, is connected by a light inextensible string passing over a smooth light fixed pulley to a smooth pulley of mass 5 kg. Two particles of masses 1 kg and 3 kg are connected by a light inextensible string passing over the 2 kg pulley. Two particles of masses 4 kg and 6 kg are connected by a light inextensible string passing over the 5 kg pulley. Find the tension in each string, when the system is released from rest.

24 N

21. 9 N

73 N

4. 8 ms

3

2

1

2

3 1 3 2

1 2

2 2

2

3 3

3



T

T

T

a

g a g a g a

T g T a T T g a

T T g a

T g b a g T a

g T b a

T g c a g T a

g T c a

3g

4g 6g

T 2

T 3

T 1

T 1

g

5g 2g

5 (b) A smooth sphere A, of mass m , moving with speed u , collides with an identical smooth sphere B which is at rest. The direction of motion of A before and after impact makes angles α and β respectively with the line of centres at the instant of impact.

The coefficient of restitution between the spheres is e.

(i) Iftan   k tan , find k , in terms of e. (ii) If the magnitude of the impulse imparted to each sphere due to the

collision is cos  8

mu , find the value of e.

cos 1 2

cos 8

(ii) 0

2 tan tan

2 tan

cos 1

2 sin

sin tan

cos 1

cos 1

NEL cos 0

(i) PCM cos 0

2

1

2

1

1 2

1 2

e

mu mu e

I mv m

e k

e k

e

k

e

u e

u

v

u

u e v

u e v

v v e u

mu m mv mv

A B

6. (a) The distance, x , of a particle from a fixed point, O , is given by

x  a sin   t 

where a , and arepositiveconstants.

(i) Show that the motion of the particle is simple harmonic.

A particle moving with simple harmonic motion starts from a point 1 m from the centre of the motion with a speed of 9.6 m s^1 and an acceleration of 16 m s^2.

(ii) Calculate a , and .

2. 6 m sin 0. 395

1. 395 rad 12

tan

cos

sin

1 sin

sin

cos 2. 4

9.6 a 4 cos

cos

4 rads

(ii)

sin

cos

(i) sin

1

2

2

2

2



a

a

a

a

x a t

a

x a t

x x

x

x a t

x a t

x a t

7. (a) A particle of mass 24 kg is attached to two light elastic strings, each of natural length 33 cm and elastic constant k.

The other ends of the strings are attached to two points on the same horizontal level 64 cm apart.

Each string makes an angle  with the

horizontal, where 4

tan  .

(i) Show that the extension of each string is 7 cm. (ii) Find the value of k.

2800 Nm^1

(ii) 2 sin 24

7 cm

(i) os

 ^ 

k

g k

T kx

T g

T g

T g

x

x

x

c

24 kg

24 g

T T

7. (b) A uniform rod BC , of length 2 p and weight W , rests in equilibrium with B in contact with a rough vertical wall. One end of a light inextensible string is fixed to a point A on the wall vertically above B , the other end is attached to C.

The coefficient of friction between the rod and the wall is .

If  CAB  BCA  θ , prove that   tan .

tan

tan

sin sin tan

cos

1 cos cos cos

sin

cos

sin cos

cos

sin

cos

sin 2 2 sin cos

sin 2 sin 2

2

T

T T

R T W

R T

T W

T p W p

T p W p





A

B

C





A

B

C

μR T

R

W

8. (b) A square lamina PQRS , of side 60 cm and mass m , can turn freely about a horizontal axis through P perpendicular to the plane of the lamina. The lamina is released from rest when PS is horizontal. (i) Find the angular velocity of the lamina when PR is vertical.

A mass m is attached to the lamina at R. The compound pendulum is set in motion. (ii) Find the period of small oscillations of the compound pendulum and hence, or otherwise, find the length of the equivalent simple pendulum.

        

   

   

1. 75 m
2. 9 2

1. 74 s

(ii) 0. 3 0. 6 2

3. 19 rads

(i) GaininKE LossinPE

2 2 3

8

1

2

(^22) 3

(^24) 3

4 2

1

2 2

1



L

g g

L

mg

m

Mgh

I

T

mg

Mgh mg mg

m

I m m

g

m m mg

I mgh

P

Q

S

R

P

Q

S

R

9. (a) A U-tube of cross-sectional area of 0.15 cm^2 contains oil of relative density 0.8.

The surface of the oil is 12 cm from the top of both branches of the U-tube.

What volume of water can be poured into one of the branches before the oil overflows in the other branch?

 

6 3

4

2. 88 10 m

Volume

1. 192 m





h A

h

gh g^5

h