1202630133 1990 mathematics extension 2 hsc

4 UNIT MATHEMATICS MECHANICS HSC

BOARD OF STUDIES NSW 1984 - 1997

EDUDATA: DATAVER1.0 1995

Mechanics 4U97-5b)!

mg

N

F

A particle of mass m is lying on an inclined plane and does not move. The plane is at an angle to the horizontal. The particle is subject to a gravitational force mg, a normal reaction force N, and a

frictional force F parallel to the plane, as shown in the diagram. Resolve the forces acting on the

particle, and hence find an expression for F

N in terms of .

tan 4U97-6b)!

A ball of mass 2 kilograms is thrown vertically upward from the origin with an initial speed of

8 metres per second. The ball is subject to a downward gravitational force of 20 newtons and air

resistance of (v2/5) newtons in the opposite direction to the velocity, v metres per second. Hence,

until the ball reaches its highest point, the equation of motion is y =v2

10

10 , where y metres is its

height.

i. Using the fact that y = vdv

dy, show that, while the ball is rising, v

2 = 164e

-y/5 - 100.

ii. Hence find the maximum height reached.

iii. Using the fact that y =dv

dt, find how long the ball takes to reach this maximum height.

iv. How fast is the ball travelling when it returns to the origin?

i) Proof ii) 2.47m iii) 0.675sec. iv) 6.25m/s 4U96-6b)!

A circular drum is rotating with uniform angular velocity round a horizontal axis. A particle P is

rotating in a vertical circle, without slipping, on the inside of the drum.

The radius of the drum is r metres and its angular velocity is radians/second. Acceleration due to gravity is g metres/second

2, and the mass of P is m kilograms.

The centre of the drum is O, and OP makes and angle to the horizontal. The drum exerts a normal force N on P, as well as a frictional force F, acting tangentially to the drum,

as shown in the diagram.




F

N

P

mgO

By resolving forces perpendicular to and parallel to OP, find an expression for F

N in terms of the

data.

g

r g

cos

sin

2

4U96-7a)!

A particle is moving along the x axis. Its acceleration is given by d x

dt

x

x

2

2 3

5 2

and the particle starts

from rest at the point x = 1.

i. Show that the particle starts moving in the positive x direction.

ii. Let v be the velocity of the particle. Show that vx x

x

2 4 5 for x 1.

iii. Describe the behaviour of the velocity of the particle after the particle passes x = 5

2.

i) Proof ii) Proof iii) For x > 2.5, v decreases tending to 1 as x tends to . 4U95-8c)!

mg

RC

FN

0 A particle of mass m travels at constant speed v round a circular track of radius R, centre C. The track

is banked inwards at an angle , and the particle does not move up or down the bank. The reaction exerted by the track on the particle has a normal component N, and a component F due

to friction, directed up or down the bank. The force F lies in the range from -N to N, where is a positive constant and N is the normal component; the sign of F is positive when F is directed up the

bank. The acceleration due to gravity is g. The acceleration related to the circular motion is of

magnitude 2

v

R, and is directed towards the centre of the track.

i. By resolving forces horizontally and vertically, show that v

Rg

Nsin Fcos

Ncos + Fsin

2

.




ii. Show that the maximum speed vmax at which the particle can travel without slipping up the

track is given by v

Rg

tan

1 tan max

2

.

[You may suppose that tan < 1.] iii. Show that if tan , then the particle will not slide down the track, regardless of its

speed.

Proof 4U94-4b)!

Alex decides to go bungee jumping. This involves being tied to a bridge at a point O by an elastic

cable of length L metres, and then falling vertically from rest from this point. After Alex free-falls L

metres, she is slowed down by the cable, which exerts a force, in newtons, of Mgk times the distance

greater than L that she has fallen (where M is her mass in kilograms, g m s/ 2 is the constant acceleration due to gravity, and k is a constant). Let x m be the distance Alex has fallen, and let v m/s

be her speed at x. You may assume that her acceleration is given by d

dxv

1

22

.

i. Show that d

dxv

1

22

= g when x L , and

d

dxv

1

22

= g - gk(x - L) when x > L.

ii. Show that v 2gL2 when Alex first passes x = L.

iii. Show that v gx kg x L2 22 ( ) for x > L.

iv. Show that Alexs fall is halted first at x L1

k

2L

k

1

k

2

.

v. Suppose 1

k

L

4 .

Show that O must be at least 2L metres above any obstruction on Alexs path.

Proof 4U93-5a)!

P

O

L

s

F

The diagram shows a simple pendulum consisting of a particle P, of mass m kg, which is attached to a

fixed point F by a string of length L metres.

The particle P moves along a circular arc in a fixed vertical plane through F.

The point O is the lowest point of the arc, OFP = , and the arc length OP = s metres. The time t is measured in seconds and g m/s in the constant gravitational acceleration.




i. Show that the tangential acceleration of P is given by d sdt

Ld

d

2

212

2

, where

d

dt.

ii. Show that the equation of the motion of the pendulum is L dd

g sin 122

.

iii. Suppose that the pendulum is given an initial angular velocity of g

L radians/second at

= 0. Show that 12 L 2 g cos 12 . Hence deduce that the maximum value of attained by the pendulum is

3

.

iv. Suppose that on the initial upward swing the angular velocity is better approximated from the

equation 12 L g cos 12g

10(2sin )2 . Use one application of Newton's method to

find the maximum value of attained by the pendulum. Take

3

as the first

approximation.

(i) (ii) (iii) Proof (iv) = 0.9681 (to 4 d.p) 4U92-5b)!

An object is fired vertically upwards with initial speed 400 m/s from the surface of the Earth. Assume

that the acceleration due to gravity at height x above the Earths surface is 10

(1 x

R)

m/ s2

2

where the radius of the Earth, R 64 106. m.

i. Show that ddx

12

v10

1xR

22

where v is the speed of the object at height x. (Neglect air resistance).

ii. Calculate the maximum height the object reaches. Give your answer to the nearest metre.

(i) Proof (ii) 8010 metres 4U92-6a)!

a Sx

a

T

ON

The diagram shows a model train T that is moving around a circular track, centre O and radius a

metres. The train is travelling at a constant speed of u m/s. The point N is in the same plane as the

track and is x metres from the nearest point on the track. The line NO produced meets the track at S.

Let TNS and TOS as in the diagram.




i. Express d

dt

in terms of a and u.

ii. Show that a x asin( ) ( )sin 0 and deduce that

d

dt

u

x a a

cos( )

( )cos cos( ).

iii. Show that d

dt

0 when NT is tangential to the track.

iv. Suppose that x = a. Show that the trains angular velocity about N when

2

is 3

5 times

the angular velocity about N when 0.

(i) d

dt

u

a

(ii) Proof (iii) Proof (iv) Proof

4U91-6b)!

When a jet aircraft touches down, two different retarding forces combine to bring it to rest. If the

aircraft has mass M kg and speed v m/s there is a constant frictional force of 1

4M newtons and a

force of 1

108Mv2 newtons due to the reverse thrust of the engines. The reverse thrust of the engines

does not take effect until 20 seconds after touchdown.

Let x be the distance in metres of the jet from its point of touchdown and let t be the time in seconds

after touchdown.

i. Show that d

2x

dt2

1

4 for 0 < t < 20 and that for t > 20, and until the jet stops,

d2x

dt2

1

108(27 v2).

ii. Prove the identity d

2x

dt2

dv

dt v

dv

dx.

iii. If the jets speed at touchdown is 60 m/s show that v = 55 and x = 1150 at the instant the reverse thrust of the engines takes effect.

iv. Show that when t > 20, x 1150 54 ln 27 (55) ln 27 v2 2 . v. Calculate how far from the touchdown point the jet comes to rest. Give your answer to the

nearest metre.

(i) Proof (ii) Proof (iii) Proof (iv) Proof (v) 1405 metres 4U90-7a)!

A mass of m kilograms falls from a stationary balloon at height h metres above the ground. It

experiences air resistance during its fall equal to mkv2, where v is its speed in metres per second and

k is a positive constant. Let x be the distance in metres of the mass from the balloon, measured

positively as it falls.

i. Show that the equation of motion of the mass is x g kv 2 , where g is the acceleration due to gravity.

ii. Find v2 as a function of x. Hint: x

d

dxv v

dv

dx

1

2

2.

iii. Find the velocity V as the mass hits the ground in terms of g, k and h.

iv. Find the velocity of the mass as it hits the ground if air resistance is neglected.

(i) Proof (ii) vg

k(1 e )2 2kx (iii) v g

ke mskh 1 2 1 (iv) v 2gh ms-1




4U89-7a)!

0P

Q

A particle of mass m kg moves in a horizontal circle with centre O and radius r metres, with uniform

speed v metres per second.

At time t, the particle is at point P while at time t + t, it is at point Q with POQ = as in the diagram.

i. Calculate the component of the velocity at Q in the direction PO.

ii. Hence show that the particle is subject to a force of mv

2

r newtons directed towards O.

iii. Suppose that the circle lies on a track banked at and angle to the horizontal as in the figure below.

0 P

r metres

Draw a diagram of all forces on the moving particle P and show that the resultant of these

forces is normal to the track precisely when tan v

2

rg. Here g ms

-2 is the acceleration due

to gravity.

(i) v sin (ii) Proof (iii)

F

N

mg

P

4U88-6)!

A body of mass one kilogram is projected vertically upwards from the ground at a speed of 20 metres

per second. The particle is under the effect of both gravity and a resistance which, at any time, has a

magnitude of 1

40v2 , where v is the magnitude of the particle's velocity at that time.

In the following questions take the acceleration due to gravity to be 10 metres per second per second.

a. While the body is travelling upwards the equation of motion is x (101

40v )2 .

i. Taking x vdv

dx , calculate the greatest height reached by the particle.

ii. Taking xdv

dt , calculate the time taken to reach this greatest height.




b. Having reached its greatest height the particle falls to its starting point. The particle is still

under the effect of both gravity and a resistance which, at any time, has a magnitude of 1

40v

2.

i. Write down the equation of motion of the particle as it falls.

ii. Find the speed of the particle when it returns to its starting point.

(a)(i) 20loge 2 metres (ii)

2 seconds (b)(i) x g v140

2 (ii) 10 2 ms-1

4U87-6) A particle of unit mass moves in a straight line against a resistance numerically equal to v + v

3, where

v is its velocity. Initially the particle is at the origin and is travelling with velocity Q, where Q > 0.

a. Show that v is related to the displacement x by the formula x tanQ v

1 Qv

1

.

b. Show that the time t which has elapsed when the particle is travelling with velocity v is given

by t1

2log

Q (1 v )

v (1 Q )e

2 2

2 2

.

c. Find v2 as a function of t.

d. Find the limiting values of v and x as t approaches .

(a) Proof (b) Proof (c)

vQ

1 Q e Q

22

2 2t 2

(d) V approaches O and x approaches tan

-1 Q.

4U86-5ii) A particle travels in a straight line away from a fixed wall. At time t secs its acceleration a ms

-2 is

given by a = t sin t. Determine the velocity v ms-1

, and the distance from the wall x metres, as

functions of t, given that v = V and x = 0, at t = 0.

v = -t cos t + sin t + V, x = -t sin t - 2cos t + Vt + 2

4U86-6) a. A cyclist is travelling with constant speed v metres per second around a circular track of

radius r metres. Draw a diagram of the forces on the cyclist and show that the cyclist must

lean inwards towards the centre of the circle at an angle , where tan = v

rg

2

and is the

angle to the vertical of the line from the mass centre to the point of wheel contact.

P B

A

OQ

G

50 metres

b. The figure represents the front view of a cyclist PQ, with mass centre G, riding on a circular

competition track AB which is banked at an angle to the horizontal. The track is so designed that when the cyclist is travelling at a constant speed of 40 km per hour around a

horizontal circle of centre O and radius OQ = 50 metres, then PQ and AB are perpendicular.

Assuming the value of g to be 10 ms-2

, show that = 14, to the nearest degree. c. Also calculate the force of the wheels on the track at Q in the direction AB, given that a

cyclist of total mass 80kg is travelling around the same circle at 50 km per hour.




(a)

m2vr

F

G

R N

mg

where N is the normal reaction of the road, F is the

frictional force and R is the resultant force of N and F. (b) Proof (c) 108N

4U85-6) A particle of mass 10kg is found to experience a resistive force, in newtons, of one-tenth of the square

of its velocity in metres per second, when it moves through the air.

The particle is projected vertically upwards from a point O with a velocity of u metres per second, and

the point A, vertically upwards from a point O with a velocity of u metres per second, and the point A,

vertically above O, is the highest point reached by the particle before it starts to fall to the ground

again. Assuming the value of g is 10 ms-2

,

a. find the time the particle takes to reach A from O;

b. show that the height OA is 50loge(1 + 10-3

u2) metres;

c. show that the particles velocity w ms-1 when it reaches O again is given by w2 = u2(1 + 10-3u

2)-1

.

(a) 10tanu

10 10

1

seconds (b) Proof (c) Proof

4U84-6) Two stones are thrown simultaneously from the same point in the same direction and with the same

non-zero angle of projection (upward inclination to the horizontal), , but with different velocities U, V metres per second (U < V). The slower stone hits the ground at a point P on the same level as the

point of projection. At that instant the faster stone just clears a wall of height h metres above the level

of projection and its (downward) path makes and angle with the horizontal. a. Show that, while both stones are in flight, the line joining them has an inclination to the

horizontal distance from P to the foot of the wall in terms of h, .

b. Show that V(tan + tan) = 2Utan, and deduce that, if = 1

2 , then U <

3

4V.

Proof

4 UNIT MATHEMATICS POLYNOMIALS HSC



Polynomials 4U97-5c)!

Suppose that b and d are real numbers and d 0 . Consider the polynomial P(z) = z4 + bz2 + d. The

polynomial has a double root . i. Prove that P(z) is an odd function. ii. Prove that is also a double root of P(z).

iii. Prove that d =b2

4.

iv. For what values of b does P(z) have a double root equal to 3i ? v. For what values of b does P(z) have real roots?

i) ii) iii) Proof iv) b = -6 v) b 0 4U96-5b)!

Consider the polynomial equation x ax bx cx d4 3 2 0 , where a, b, c, and d are all integers. Suppose the equation has a root of the form ki, where k is real, and k 0.

i. State why the conjugate -ki is also a root.

ii. Show that c k a 2 .

iii. Show that c a d abc2 2 . iv. If 2 is also a root of the equation, and b = 0, show that c is even.

Proof 4U95-5b)!

Let f(t) = t3 + ct + d, where c and d are constants.

Suppose that the equation f(t) = 0 has three distinct real roots, t1, t2, t3.

i. Find t1 + t2 + t3.

ii. Show that t1 + t2 + t3 = - 2c.

iii. Since the roots are real and distinct, the graph of y = f(t) has two turning points, at t = u and

t = v, and f(u).f(v) < 0. Show that 27d2 + 4c

3 < 0.

(i) 0 (ii) Proof (iii) Proof 4U94-8b)!

Let x = be a root of the quartic polynomial P x x Ax Bx Ax( ) 4 3 2 1, where A and B are

real. Note that may be complex.

i. Show that 0.

ii. Show that x = is also a root of Q x xx

A xx

B( )

2 2

1 1.

iii. With u x1x

, show that Q(x) becomes R u u Au B( ) ( ) 2 2 .

iv. For certain values of A and B, P(x) has no real roots. Let D be the region of the AB plane

where P(x) has no real roots for A 0.




-2

0

L

B

T

c

A

The region D is shaded in the figure. Specify the bounding straight-line segment L and curved

segment c. Determine the coordinates of T.

(i) Proof (ii) Proof (iii) Proof (iv) The straight line segment is B = 2A - 2 and the curved segment is

B A 2142 . T is the point (4, 6).

4U93-1d)!

i. Find real numbers a, b and c such that 4x 3

(x 1)(x 2)

ax b

x 1

c

x 22 2

ii. Hence find 4x 3

(x 1)(x 2) dx2

.

(i) a = 1, b = 2, c = -1 (ii) 12 ln 2tan x ln c1x x2 1 2




4U93-5b)!

i. If a is a multiple root of the polynomial equation P(x) = 0, prove that P(a) = 0. ii. Find all of the roots of the equation 18x

3 + 3x - 28x + 12 = 0, given that two of the roots are

equal.

(i) Proof (ii) x 23,23

32 ,

4U91-7b)!

Let x = a be a root of the quartic polynomial P(x) = x4 + Ax

3 + Bx

2 + Ax + 1 where (2 + B)

2 4A2.

i. Show that cannot be 0, 1, or -1.

ii. Show that x 1

is a root.

iii. Deduce that if is a multiple root, then its multiplicity is 2 and 4B = 8 + A2.

Proof 4U90-6a)!

i. Write down the relations which hold between the roots , , of the equation

ax bx cx d a3 2 0 0 , ( ) , and the coefficients a, b, c, d.

ii. Consider the equation 36x3 12x2 11x 2 0 . You are given that the roots ,, of this

equation satisfy . Use part (i) to find .

iii. Suppose the equation x3 px2 qx r 0 has roots , , which satisfy .

Show that p3 4pq8r 0.

(i) b

a,

c

a,

d

a (ii)

1

6 (iii) Proof

4U89-2b)!

i. Write 4x2 5x7

(x 1)(x2 x2) in the form

A

x 1

BxC

x2 x 2

.

ii. Hence evaluate 4x2 5x7

(x1)(x2 x2)dx

1

0

.

(i)

2

x 1

6x 3

x x 22 (ii) 2 ln 2

4U87-8ii)

a. A polynomial R(x) is given by R(x) = x7 - 1. Let p 1 be that complex root of R(x) = 0

which has the smallest positive argument. Show that:

. R(x) = (x - 1)(1 + x + x2 + x3 + x4 + x5 + x6)

. 1 + p + p2 + p3 + p4 + p5 + p6 = 0

b. Let = p + p2 + p4 and = p3 + p5 + p6.

. Prove that + = -1 and = 2.

. Show that 1 7

2i

and 1 7

2i

c. Given that

T(x) = 1 + x + x2 + x

3 + x

4 + x

5 + x

6

= (x - p)(x - p2) (x - p

3) (x - p

4) (x - p

5) (x - p

6),

write the polynomial T(x) as a product of two cubics with coefficients involving , and rational numbers.

(a) Proof (b) Proof (c) (x p)(x p )(x p ) (x x x 1)(x x x2 4 3 2 3 2 1)

4U86-3ii) a. Show that if a is a multiple root of the polynomial equation f(x) = 0 then f(a) = f '(a) = 0.

b. The polynomial xn+1 + xn + 1 is divisible by (x - 1)2. Show that = n, and = -(1 + n).




c. Prove that 1 + x + x

2

2! ...

xn

n! has no multiple roots for any n 1.

Proof

4U86-8i)

Let , , be the roots of the cubic equation x3 + px2 + q = 0, where p, q are real. The equation

x3 + ax

2 + bx + c = 0 has roots 2, 2, 2. Find a, b, c as functions of p, q.

a = -p2, b = -2pq and c = -q2

4U85-1ii)

Find real numbers A, B, C such that x

(x 1)2(x2)

A

x 1

B

(x1)2

C

x 2.

Hence show that x

(x 1) (x 2)dx 2log

3

21

2

0

1

2

e

.

A = -2, B = -1, C = 2, Proof

1202630133 1990 mathematics extension 2 hsc

Documents