Higher Level Applied Mathematics Examination Questions
Take the value of g to be 9.8 m/s2.
2005
1. (a) Car
A and car B travel in the same direction along a horizontal straight road.
Each car is travelling at a uniform speed of 20 m/s.
Car A is at a distance of d metres in front of car B.
At a certain instant car A starts to brake with a constant retardation of
6 m/s2.
0.5 s later car B starts to brake with a constant retardation of 3 m/s2 .
Find
(i) the distance travelled by car A before it comes to rest
(ii) the minimum value of d for car B not
to collide with car A.
Answer
(b) A
mass of 8 kg falls freely from rest. After 5 s the mass penetrates sand.
The sand offers a constant resistance and brings the mass to rest in 0.01
s.
Find
(i) the constant resistance of the sand
(ii) the distance the mass penetrates into the sand.
Answer
2004
1 (a). A ball is
thrown vertically upwards with an initial velocity of 20 m/s.
Once second later, another ball is thrown vertically upwards from the same
point with an initial velocity of u m/s.
The balls collide after a further 2 seconds.
(i). Show that u = 17.75.
(ii).
Find the distance traveled by each ball before the collision,
giving your answers correct to the nearest metre.
Answer
2003
1 (a). The points
p, q and r all lie in a straight line.
A train passes point p with speed u m/s. The train is traveling
with uniform retardation f m/s2. The train takes 10 seconds
to travel from p to q and 15 seconds to travel from q
to r, where |pq|=|qr|=125 metres.
(i). Show that f = 1/3.
(ii). The train comes to rest s metres after passing r.
Find s, giving your answer correct to the nearest metre.
Answer
2003 1(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.
Answer
2002
1. (a) A stone is thrown vertically
upwards under gravity with a speed of u m/s from a point 30 metres
above the horizontal ground.
The stone hits the ground 5 seconds later.
(i) Find the value of u..
(ii) Find the speed with which the stone hits the ground.
(b) A particle, with initial speed u, moves in a straight line
with constant acceleration.
During the time interval from 0 to t, the particle travels a distance
p .
During the time interval from t to 2t, the particle travels
a distance q.
During the time interval from 2t to 3t, the particle travels
a distance r.
(i) Show that 2q = p
+ r.
(ii) Show that the particle travels
a further distance 
in the time interval
from 3t to 4t.
2001
1. (a) Points p and q
lie in a straight line, where |pq| = 1200 metres.
Starting from rest at p, a train accelerates at 1
m/s2 until it reaches the speed limit of 20 m/s. It continues at
this speed of 20 m/s and then decelerates at 2 m/s2, coming to
rest at q.
(i) Find the time it takes the train to go from p
to q
(ii)
Find the shortest time it takes the train to go from rest at p
to rest at q if there is no speed limit, assuming that the acceleration
and deceleration remain unchanged at 1 m/s2 and 2 m/s2,
respectively.
(b) A particle is projected vertically
upwards with an initial velocity of u m/s and another particle is
projected vertically upwards from the same point and with the same initial
velocity T seconds
later.
Show that the particles
(i) will meet 
seconds from
the instant of projection of the first particle
(ii) will meet at a height of
metres.
2000
1. (a) A stone projected vertically upwards
with an initial speed of u m/s rises 70 m in the first t seconds
and another 50 m in the next t seconds.
Find the value of u.
(b) A car, starting from rest and travelling from p to q
on a straight level road,
where | pq |
= 10 000 m, reaches its maximum speed 25 m/s by constant acceleration
in the first 500 m and continues at this maximum speed for the rest of the
journey.
A second car, starting from rest and travelling from q to p,
reaches the same maximum speed by constant acceleration in the first 250 m
and continues at this maximum speed for the rest of the journey.
(i) If the two cars start at the same time, after how many seconds
do the two cars meet ?
Find, also, the distance travelled by each car
in that time.
(ii) If the start of one car is delayed so that they meet each other
exactly halfway between p and q, find which car is delayed and
by how many seconds.
1999
1 (a). A car of mass 1500 kg
travels up a slope of gradient 
against a
constant resistance of 0.2 N per kilogram. Find
(i). the constant force required to produce an acceleration of 0.1 m/s2.
(ii).
the power which is developed when the speed is 20 m/s.
(b)
A particle travels in a straight line with constant acceleration f
for 2t seconds and covers 15 metres. The particle then travels a further
55 metres at constant speed in 5t seconds. Finally the particle is
brought to rest by a constant retardation of 3f.
(i). Draw a speed-time graph for the motion of the particle.
(ii). Find the initial velocity in of the particle in terms of t.
(iii). Find the total distance travelled in metres, correct to two decimal places.
1998
1 (a). A train accelerates
uniformly from rest to a speed v m/s. It continues at this constant
speed for a period of time and then decelerates uniformly to rest. If the
average speed for the whole journey is 
, find what
fraction of the whole distance is described at constant speed.
(b) Car A, moving with uniform acceleration
m/s2
passes a point p with speed 9u m/s. Three seconds later car
B, moving with uniform acceleration 
m/s2
passes the same point with speed 5u m/s. B overtakes A when their speeds
are 6.5 m/s and 5.4 m/s respectively. Find
(i). the value u and the value b
(ii). the distance travelled from p until overtaking occurs.