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This is the Exam of Applied Mathematics which includes Vertically Upwards, Initial Velocity, Balls Collide, Distance Travelled, Nearest Metre, Horizontal Road, Friction, Air Resistance, Caravan etc. Key important points are:Slips, Car Travels, Straight Level Road, Decelerates Uniformly, Acceleration, Deceleration, Distance, Speed, Positioned, Constant Speed
Typology: Exams
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1. A car travels along a straight level road. It passes a point P with a speed of 8 m s^1 and accelerates uniformly for 12 seconds to a speed of 32 m s^1. It then travels at a constant speed of 32 m s^1 for 7 seconds. Finally the car decelerates uniformly from 32 (^) m s^1 to rest at a point Q. The car travels 128 metres while decelerating. Find (i) the acceleration
(ii) the deceleration (iii) | PQ |, the distance from P to Q
(iv) the speed of the car when it is 72 m from Q.
24 m s
(iv) 2
592 m
128 m
224 m
240 m.
(iii)
4 ms
(ii) 2
2 ms
(i)
1
(^22)
2 2
3
2
1
1 21
2 21
2
2 2
2 2
2
u
u
v u fs
s
s
s
s
s ut at
f
f
v u fs
f
f
v u ft
2. Ship A is positioned 80 km south of ship B. A is moving north-east at a constant speed of 30 2 km h^1.
B is moving due west at a constant speed of 15 kmh^1.
Find (i) the velocity of A in terms of
i and^
j (ii) the velocity of B in terms of
i and^
j (iii) the velocity of A relative to B in terms of
i and
j (iv) the shortest distance between A and B in the subsequent motion.
A
B
AB A B
(i) V 30 2 in45 i 30 2cos45 j 30 i 30 j
(ii) V 15 i 0 j
(iii) V V V 30 i 30 j 15 i 0 j 45 i 30 j
(iv)
s
tan 1 30 45 33·
80cos33· 66·56 km
d
15 km h–1^ B
80 km 30 2 km h^
45°
80 km
θ
d
3 (b) A particle is projected with initial velocity 21
i + 50
j m s^1 from point P on a horizontal plane. A and B are two points on the trajectory (path) of the particle.
The particle reaches point A after 3 seconds of motion.
The displacement of point B from P is k
i + 80
j metres. Find (i) the velocity of the particle at A in terms of
i and
j (ii) the speed and direction of the particle at A (iii) the value of k.
2 2 1
1
2
(i) 21 0 21
50 10 3 20
(ii) 21 20 29 m s
tan 20 21 43·
(iii) 80 50 5
x
y
v u at v
v
v i j
v
t t
2
1 2 2
x
t t t t t
s ut at k
21 m s –
50 m s –
80 m
k m
4. (a) Two particles of masses 2 kg and 3 kg are connected by a taut, light, inextensible string which passes over a smooth light pulley. The system is released from rest.
Find (i) the common acceleration of the particles (ii) the tension in the string.
(i)
2 ms^2 5
a^ g
g a
T g a
g T a
(ii)
T g a
2 kg (^) 3 kg
5. A smooth sphere A, of mass 5 kg, collides directly with another smooth sphere B, of mass 2 kg, on a smooth horizontal table.
A and B are moving in the same direction with speeds of 4 m s^1 and 1 m s^1 respectively.
The coefficient of restitution for the collision is 1. 6 Find (i) the speed of A and the speed of B after the collision (ii) the loss in kinetic energy due to the collision (iii) the magnitude of the impulse imparted to A due to the collision.
1 2 1 2
1 2
1 1 2 1
b
(i) 5 4 2(1) 5 2 22 5 2
3 m s and 7 m s 2
(ii) KE
v v v v
v v e
v ^ v
1 2 1 2 2 2
1 2 1 2 a 2 2
b a
(iii) Impulse 5 3 5 4 5 N s
A 5 kg B 2 kg
4 m s^1 1 m s^1
6. (a) Particles of weight 4 N, 7 N, 3 N and 5 N are placed at the points
Find (i) the value of p (ii) the value of q.
(b) A triangular lamina with vertices A , B and C has the portion inside its incircle (the circle that touches the three sides of the triangle) removed. D is the centre of the incircle. The co-ordinates of the points
Find the co-ordinates of the centre of gravity of the remaining lamina.
(a) 19 3 4 2 7 1 3 5 13 19 5
(b) rea: c.g. 36 27 486 12,
p p p q q q a
22 2 7
circle 9 254·57 9, 9
lamina 231·
x, y
x x
y y
8. (a) A particle describes a horizontal circle of radius 2 metres with uniform angular velocity ω radians per second. Its speed is 6 m s^1 and its mass is 4 kg.
Find (i) the value of ω (ii) the centripetal force on the particle. (b) A hemispherical bowl of diameter 20 cm is fixed to a horizontal surface.
A smooth particle of mass 1 kg describes a horizontal circle of radius r cm on the smooth inside surface of the bowl. The plane of the circular motion is 4 cm above the horizontal surface. Find (i) the value of r (ii) the reaction force between the particle and the surface of the bowl (iii) the angular velocity of the particle. (a)
(ii)
(i)
2
2
F mr
v r
(b) 2 2
2 2
(i) 10 6 8 cm
(ii) sin 1 (^6 10) 16·7 N 10
(iii) cos (^100 8 1) 0· 6 10 12·
r
R g R R
R mr
20 cm
4 cm
9. (a) State the Principle of Archimedes. A solid piece of metal has a weight of 26 N. When it is completely immersed in water the metal weighs 21 N. Find (i) the volume of the metal (ii) the relative density of the metal.
(b) A right circular solid cylinder has a base of radius 8 cm and a height of 18 cm.
The relative density of the cylinder is 3 and it is completely immersed in a tank of liquid of relative density 0·9. The cylinder is held at rest by a light inextensible vertical string which is attached to a fixed point P. The upper surface of the cylinder is horizontal. Find the tension in the string. [ Density of water = 1000 kg m^3 ].
(a)
3
Principle of Archimedes
(i) 5 1000 10 0·0005 m
(ii) 26 26 26 1000 0·0005 10 5·
B Vg V V
W Vg s s
(b) ^ ^
^ ^ ^ ^
2
2
