CAIE M1 (Mechanics 1) 2022 November

1 A particle \(P\) is projected vertically upwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on the ground. \(P\) reaches its greatest height after 3 s .

1. Find \(u\).
2. Find the greatest height of \(P\) above the ground.

2 A box of mass 5 kg is pulled at a constant speed of \(1.8 \mathrm {~ms} ^ { - 1 }\) for 15 s up a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The box moves along a line of greatest slope against a frictional force of 40 N . The force pulling the box is parallel to the line of greatest slope.

1. Find the change in gravitational potential energy of the box.
2. Find the work done by the pulling force.

3
\includegraphics[max width=\textwidth, alt={}, center]{4a2bad7c-6720-414c-b336-060afb2255e9-05_610_591_257_778} A ring of mass 4 kg is threaded on a smooth circular rigid wire with centre \(C\). The wire is fixed in a vertical plane and the ring is kept at rest by a light string connected to \(A\), the highest point of the circle. The string makes an angle of \(25 ^ { \circ }\) to the vertical (see diagram). Find the tension in the string and the magnitude of the normal reaction of the wire on the ring.

4 A particle \(P\) travels in the positive direction along a straight line with constant acceleration. \(P\) travels a distance of 52 m during the 2 nd second of its motion and a distance of 64 m during the 4th second of its motion.

1. Find the initial speed and the acceleration of \(P\).
2. Find the distance travelled by \(P\) during the first 10 seconds of its motion.

5 Particles \(X\) and \(Y\) move in a straight line through points \(A\) and \(B\). Particle \(X\) starts from rest at \(A\) and moves towards \(B\). At the same instant, \(Y\) starts from rest at \(B\). At time \(t\) seconds after the particles start moving

• the acceleration of \(X\) in the direction \(A B\) is given by \(( 12 t + 12 ) \mathrm { m } \mathrm { s } ^ { - 2 }\),
• the acceleration of \(Y\) in the direction \(A B\) is given by \(( 24 t - 8 ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  1. It is given that the velocities of \(X\) and \(Y\) are equal when they collide.

Calculate the distance \(A B\).

It is given instead that \(A B = 36 \mathrm {~m}\). Verify that \(X\) and \(Y\) collide after 3 s.

6 A car of mass 1750 kg is pulling a caravan of mass 500 kg . The car and the caravan are connected by a light rigid tow-bar. The resistances to the motion of the car and caravan are 650 N and 150 N respectively.

1. The car and caravan are moving along a straight horizontal road at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Find the power of the car's engine.
   2. The engine's power is now suddenly increased to 40 kW . Find the instantaneous acceleration of the car and caravan and find the tension in the tow-bar.
2. The car and caravan now travel up a straight hill, inclined at an angle \(\sin ^ { - 1 } 0.14\) to the horizontal, at a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car's engine is working at 31 kW . The resistances to the motion of the car and caravan are unchanged. Find \(v\).

7
\includegraphics[max width=\textwidth, alt={}, center]{4a2bad7c-6720-414c-b336-060afb2255e9-12_560_716_258_712} Particles of masses 1.5 kg and 3 kg lie on a plane which is inclined at an angle of \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The section of the plane from \(A\) to \(B\) is smooth and the section of the plane from \(B\) to \(C\) is rough. The 1.5 kg particle is held at rest at \(A\) and the 3 kg particle is in limiting equilibrium at \(B\). The distance \(A B\) is \(x \mathrm {~m}\) and the distance \(B C\) is 4 m (see diagram).

1. Show that the coefficient of friction between the particle at \(B\) and the plane is 0.75 .
   The 1.5 kg particle is released from rest. In the subsequent motion the two particles collide and coalesce. The time taken for the combined particle to travel from \(B\) to \(C\) is 2 s . The coefficient of friction between the combined particle and the plane is still 0.75 .
2. Find \(x\).
3. Find the total loss of energy of the particles from the time the 1.5 kg particle is released until the combined particle reaches \(C\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.