CAIE M1 (Mechanics 1) 2020 March

1 A lorry of mass 16000 kg is travelling along a straight horizontal road. The engine of the lorry is working at constant power. The work done by the driving force in 10 s is 750000 J .

1. Find the power of the lorry's engine.
2. There is a constant resistance force acting on the lorry of magnitude 2400 N . Find the acceleration of the lorry at an instant when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2 A particle \(P\) of mass 0.4 kg is on a rough horizontal floor. The coefficient of friction between \(P\) and the floor is \(\mu\). A force of magnitude 3 N is applied to \(P\) upwards at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The particle is initially at rest and accelerates at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the time it takes for \(P\) to travel a distance of 1.44 m from its starting point.
2. Find \(\mu\).

3
\includegraphics[max width=\textwidth, alt={}, center]{9ac08732-e825-473a-943c-8ad8e9e0bc17-04_519_1018_260_561} The diagram shows the vertical cross-section of a surface. \(A , B\) and \(C\) are three points on the crosssection. The level of \(B\) is \(h \mathrm {~m}\) above the level of \(A\). The level of \(C\) is 0.5 m below the level of \(A\). A particle of mass 0.2 kg is projected up the slope from \(A\) with initial speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle remains in contact with the surface as it travels from \(A\) to \(C\).

1. Given that the particle reaches \(B\) with a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and that there is no resistance force, find \(h\).
2. It is given instead that there is a resistance force and that the particle does 3.1 J of work against the resistance force as it travels from \(A\) to \(C\). Find the speed of the particle when it reaches \(C\).

4 A cyclist travels along a straight road with constant acceleration. He passes through points \(A , B\) and \(C\). The cyclist takes 2 seconds to travel along each of the sections \(A B\) and \(B C\) and passes through \(B\) with speed \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance \(A B\) is \(\frac { 4 } { 5 }\) of the distance \(B C\).

1. Find the acceleration of the cyclist.
2. Find \(A C\).

5
\includegraphics[max width=\textwidth, alt={}, center]{9ac08732-e825-473a-943c-8ad8e9e0bc17-08_572_572_262_790} Coplanar forces, of magnitudes \(F \mathrm {~N} , 3 \mathrm {~N} , 6 \mathrm {~N}\) and 4 N , act at a point \(P\), as shown in the diagram.

1. Given that \(\alpha = 60\), and that the resultant of the four forces is in the direction of the 3 N force, find \(F\).
2. Given instead that the four forces are in equilibrium, find the values of \(F\) and \(\alpha\).

6 On a straight horizontal test track, driverless vehicles (with no passengers) are being tested. A car of mass 1600 kg is towing a trailer of mass 700 kg along the track. The brakes are applied, resulting in a deceleration of \(12 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The braking force acts on the car only. In addition to the braking force there are constant resistance forces of 600 N on the car and of 200 N on the trailer.

1. Find the magnitude of the force in the tow-bar.
2. Find the braking force.
3. At the instant when the brakes are applied, the car has speed \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At this instant the car is 17.5 m away from a stationary van, which is directly in front of the car. Show that the car hits the van at a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. After the collision, the van starts to move with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the car and trailer continue moving in the same direction with speed \(2 \mathrm {~ms} ^ { - 1 }\). Find the mass of the van.

7 A particle moves in a straight line through the point \(O\). The displacement of the particle from \(O\) at time \(t \mathrm {~s}\) is \(s \mathrm {~m}\), where $$\begin{array} { l l } s = t ^ { 2 } - 3 t + 2 & \text { for } 0 \leqslant t \leqslant 6 ,
s = \frac { 24 } { t } - \frac { t ^ { 2 } } { 4 } + 25 & \text { for } t \geqslant 6 . \end{array}$$

1. Find the value of \(t\) when the particle is instantaneously at rest during the first 6 seconds of its motion.
   At \(t = 6\), the particle hits a barrier at a point \(P\) and rebounds.
2. Find the velocity with which the particle arrives at \(P\) and also the velocity with which the particle leaves \(P\).
3. Find the total distance travelled by the particle in the first 10 seconds of its motion.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.