Advanced level 2024 CASPA mock mathematics with mechanics 3

Advanced level 2024 CASPA mock mathematics with mechanics 3

Advanced level 2024 CASPA mock mathematics with mechanics 3

i) A particle moving in the X-Y plane has position vector r at time t seconds given by
𝒓 = (𝑐𝑜𝑠 4𝑡)𝒊 + (𝑠𝑖𝑛 4𝑡)𝒋
a) Show that the speed of the particle at any instant is 4ms-1.
b) Find the position vector of the particle when t = π
2
and when t = π
16
.
c) Show that the magnitude of the position vector is constant and find its value. (3, 2, 2 marks)
ii) Another particle moves in the X-Y plane at time t seconds. When it is at the point (x, y), it has
acceleration given by
𝑑2𝑥
𝑑𝑡2 = 8 𝑚𝑠−2, 𝑑𝑑2𝑡𝑦2 = 0𝑚𝑠−2
Initially, the particle is at the point (0,0) and its velocity is given by 𝑑𝑥
𝑑𝑡
= 0 and 𝑑𝑦
𝑑𝑡
= 𝑢 𝑚𝑠−1, where u is a
non-zero constant. Show that the particle is moving along a curve (6 marks)
2. A uniform square lamina PQST whose edges are each of length 4a and a uniform equilateral triangle
QRS are joined together to form a single composite uniform lamina PQRST
a) Find the distance from QS and the distance from PQ, of the centre of gravity of the composite lamina
The lamina PQRST is suspended freely at Q
b) Find, in degrees and to one decimal place, the angle θ which PQ makes with the vertical when the lamina
is in equilibrium. (9, 4 marks)
3. A block P of mass 60kg rests on a smooth horizontal table and is attached by light in-extensible strings to
blocks Q and R of masses 12kg and 18kg respectively. The strings pass over light smooth pulleys on opposite
edges of the table so that Q and R hang freely. The system is then released from rest with both strings taut and
the hanging parts vertical. Determine
a) The magnitude of the acceleration of the system,
b) The tensions 𝑇1 and 𝑇2 acting on the block
Block R falls a distance 4m and is brought to rest. Given that P remains on the table and the string joining Q
and P is long enough that Q cannot be stopped by the edge of the table,
c) Calculate the further distance that Q covers vertically upwards before momentarily coming to rest
( take g as 10m𝑠−2) (6, 2, 5 marks)
4. Three smooth spheres A, B, C of equal radius and masses 2m, 7m and 14m respectively, lie at rest in a
straight line on a smooth horizontal floor, with B between A and C. The coefficient of restitution between each
pair of spheres is 1
2
. Sphere A is projected with speed u to collide directly with sphere B which subsequently
collides directly with C.
a) Show that the sphere B comes to rest after its collision with C.
b) Find the total kinetic energy of the system immediately after B collides with C. (10, 3 marks)
5. A particle is projected with speed 20√5 ms-1 from the top of a cliff so that it hits a target which is 200m
horizontally from the foot of the cliff and 200m below the vertical line through the point of projection.
a) Show that there are two possible directions of projection and that these directions are at right angles.
Hence
b) Find the corresponding times of flight.
Using the angle of projection whose value lies in the first quadrant,
c) Find the maximum height above the point of projection attained by the particle, giving your answer
correct to two decimal places. ( take g as 10ms-2) (6, 4, 3 marks)