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This is the Exam of Applied Mathematics which includes Numerical Slips, Particles Collide, Distance, Projected Vertically, Downwards, Particle Falls, Initial Velocity, Distance, Value etc. Key important points are: Mark Scripts, Uniform Retardation, Metres, Passing, Nearest Metre, Accelerate Uniformly, Constant Speed, Distance, Closest Distance, Woman Travelling
Typology: Exams
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General Guidelines
1 Penalties of three types are applied to candidates' work as follows:
Slips - numerical slips S(-1)
Blunders - mathematical errors B(-3)
Misreading - if not serious M(-1)
Serious blunder or omission or misreading which oversimplifies:
Attempt marks are awarded as follows: 5 (att 2).
2 Mark all answers, including excess answers and repeated answers whether cancelled or not, and award the marks for the best answers.
3 Mark scripts in red unless candidate uses red. If a candidate uses red, mark the script in blue or black.
4 Number the grid on each script 1 to 10 in numerical order, not the order of answering.
5 Scrutinise all pages of the answer book.
6 The marking scheme shows one correct solution to each question. In many cases there are other equally valid methods.
(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 from rest 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 gets to the bus is 17.5 metres.
( ) ( )
closestdistance 40 0.1 15 315
(ii) distance 40 s
After 20 seconds 40 20 40 200
Man 0
(i) Bus 0 20
2
2
2 2
1
man
2 2
1
2 2
(^21) 2
1
t t dt
d
t t
at ut
s
u m s (and a )
v v u a
s s u a
s ut at s u
v u at v u
s ut at s a
v u at v a
bus
man bus
man bus
man
bus
2. (a) A woman travelling north at 10 km/h finds that the wind appears to blow from the west. When the woman trebles her speed, the wind appears to blow from the north-west.
Find the velocity of the wind.
20 10 km/h
V i j
a y
y b
y i y j a i b j
V a i b j
V y i y j
V j
b
x i a i b j
V a i b j
V x i
V j
W
WP W P
W
WP
P
WP W P
W
WP
P
r r r
r r r r
r r r
r r r
r r r
r r
r r r
r r r
r r r
r r
r r
3. (a) A particle is projected from a point on level horizontal ground at an
height reached by the particle.
{ ( )} { ( ) }
tan or 38. 7
2 sin cos 5 sin
Range 5 Maximumheight
sin
sin sin
sin sin Maxheight sin
sin Maxheight
2 sin cos
2 sin Range cos
2 sin
Range 0
cos sin
5
4
2 2 2
2 2
2 2 2 2
2 2
1
2
j
2 2
1
g
u g
u
g
u
g
u g
u
g
u g g
u u
g
u t
g
u
g
u u
g
u t
r
r u t i u t gt j
r
r r
(b) A particle is projected up an inclined plane with initial velocity u m/s. The line of projection makes an angle α with the horizontal and the inclined plane makes an angle β with the horizontal. (The plane of projection is vertical and contains the line of greatest slope.)
Find, in terms of u , g , α and β , the range of the particle up the inclined plane.
gcos
u
2 sin cos(2 ) gcos
u
2 sin cos cos( ) sin sin( ) gcos
u
gcos
2usin( ) sin ..
gcos
2usin( ) ucos( ).
ucos( ).t sin .t
Range
gcos
2usin( ) t
usin( ).t cos .t 0
2
2
2
2
2
2
2 2
1
2 2
1
2 2
1
g
g
r
g
r
i
j (^) 5
(b) A block of mass 4 kg rests on a rough plane inclined at 60° to the horizontal. It is connected by a light inextensible string which passes over a smooth, light, fixed pulley to a particle of mass 8 kg which hangs freely under gravity. The coefficient of friction between the block and the plane is 4
. The system starts from rest
with the block at a distance of 2 m from the pulley. The 8 kg mass moves vertically downwards.
(i) Show that the tension in the string is 52 N, correct to the nearest whole number. (ii) How far has the block moved up the plane after 1 second? (iii) After 1 second the string is cut. Determine whether or not the block will reach the pulley.
( )( )
the pulley
block willreach
As 0
3.3 2 sin 60 0. 35
(iii) Secondstage 2
(ii) Firstsecond
4 sin 60 4 cos 60 4
(i) 8 8
2
1
8
(^21)
2 2
1
2
1
2 2
1
2
4
1
−
−
−
v
ms
g g
v u as
ms
v u at
s ut at
a ms
T g g a
g T a^5
5. (a) A smooth sphere P, of mass 3 m , moving with speed u , collides directly with a smooth sphere Q, of mass 13 m , which is at rest. Sphere Q then collides with a vertical wall which is perpendicular to the direction of motion of the spheres. The coefficient of restitution for all of the collisions is e.
Find (i) the speed, in terms of u and e , of each sphere after the first collision (ii) the range of values of e for which there will be a second collision between the spheres.
0 e
(ii) Therewillbeasecondcollisionif:
NEL v v e u 0
(i) PCM 3mu 13m( 0 ) 3mv 13 mv
3
1
2
2 1
2
1
1 2
1 2
e e
e e
e
u e
u e
ev v
e
u v
e u v
about a fixed point o. The maximum speed of the particle is 8 cm/s. (i) Find the amplitude of the motion. (ii) Find the speed of the particle when it is at a distance of 3 cm from o.
2 7 cm/s
4 cm
Period
2 2
max
v
v ω a x
a
a
(b) A particle of mass m is held at a point p on
The particle is released from rest. When the particle reaches an arbitrary point q , its speed is v.
(ii) If 3
{ }
⇒ = ( ) °
cos−^ or 63.
3 cos
cos 2 cos
2 cos cos cos 0
(ii) cos
2 cos cos
cos cos
Energyat Energyat
PEat cos
PEat cos
KEat
(i) KEat 0
9
(^14)
3
4
3
2
2
2
2 2
1
2 2 1
β
β
r
m gr mg β
r
mv mg β R
v gr
mv mgr mgr
q p
q mgr
p mgr
q mv
p 5 5 5 5 5
(b) Two uniform ladders, [ ab ] and [ bc ], each of weight W and length l , are smoothly jointed at b.
rough horizontal ground.
μR μS
( ) ( ) ( )
μ tan
μ tan
W tan W tan μW
R sin W sin μR cos
Takemomentsabout forrod :
vert: R S W W
Resolveforcesactingonsystem:
Takemomentsabout forsystem:
2 1
2
1
2
1
2
1
l l l
b ba
a
8. (a) Prove that the moment of inertia of a uniform rod of mass m and length 2 l
about an axis through its centre perpendicular to the rod is 2 3
ml.
2 3
1
3 3
2
3
2
2
momentofinertiaoftherod
momentofinertiaoftheelement
massofelement
Let massperunitlength
l
l
l
l
l l
m
x M
M x dx
Mdx x
Mdx
−
∫−
9. (a) A rectangular tank is 1 m high and has a base 2 m by 3 m. It is half filled with water and is tilted about one of the shorter edges of its base until the water just begins to overflow. When the tank is tilted to this position, a diagonal of its vertical faces lies along the surface of the water. With the tank held in this position, (i) show that the depth, d , of the water in the tank is 10
m
(ii) show that the thrust on the base of the tank is 900 10 g N (iii) find the thrust on a vertical side of the tank (e.g., the side facing towards you in the diagram).
( ) ( ) ( ) ( )
( ) { }
( ) { ( ) ( )}
150 10 g
1000 g
g d 1 3
(iii) Thrust Pressure Area
900 10 g
1000 g
g d 3 2
(ii) Thrust Pressure Area
d
d 10 1 3
(i) areaoftriangle 1 3
2 1 3 1
2 1
2
1 2
1
2 1
(b) The trapezium pqrs is a cross-section of a block of wood. The distance between the parallel sides [ pq ] and [ rs ] is 20 cm. | pq | = 34 cm, | rs | = 14 cm and | ps | = | qr |.
The relative density of the wood is 10
The block floats with the face containing [ pq ] immersed in water and horizontal.
Find the depth of side [ pq ] below the surface of the water.
{ ( )}{ } { }{ }
{ ( ) }{ }
{ }{ } { }{ }
h 12 cm
3 4h h 336
10003 4h h 700 480
Weightofblockofwood W Vg
10003 4h h
1000 34 34 hh
Vg
BuoyancyB weightofliquiddisplaced
Letdepthof pq below thesurface h
2 2
1
2 2
1
2 1
2 2 1
2
1
x g x g
x g
x g
x g
x g
