CAIE M1 (Mechanics 1) 2017 March

1 A particle of mass 0.4 kg is projected with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal.

1. Find the initial kinetic energy of the particle.
2. Use an energy method to find the distance the particle moves up the plane before coming to instantaneous rest.
   \includegraphics[max width=\textwidth, alt={}, center]{f3ddfde8-b678-4c3a-a9a6-d984c2c38bd9-03_266_874_260_632} A particle \(P\) of mass 1.6 kg is suspended in equilibrium by two light inextensible strings attached to points \(A\) and \(B\). The strings make angles of \(20 ^ { \circ }\) and \(40 ^ { \circ }\) respectively with the horizontal (see diagram). Find the tensions in the two strings.
   \includegraphics[max width=\textwidth, alt={}, center]{f3ddfde8-b678-4c3a-a9a6-d984c2c38bd9-04_286_664_251_737} A particle of mass 0.6 kg is placed on a rough plane which is inclined at an angle of \(21 ^ { \circ }\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(P \mathrm {~N}\) acting parallel to a line of greatest slope of the plane, as shown in the diagram. The coefficient of friction between the particle and the plane is 0.3 . Show that the least possible value of \(P\) is 0.470 , correct to 3 significant figures, and find the greatest possible value of \(P\).

4 A car of mass 900 kg is moving on a straight horizontal road \(A B C D\). There is a constant resistance of magnitude 800 N in the sections \(A B\) and \(B C\), and a constant resistance of magnitude \(R \mathrm {~N}\) in the section \(C D\). The power of the car's engine is a constant 36 kW .

1. The car moves from \(A\) to \(B\) at a constant speed in 120 s . Find the speed of the car and the distance \(A B\).
   The car's engine is switched off at \(B\).
2. The distance \(B C\) is 450 m . Find the speed of the car at \(C\).
3. The car comes to rest at \(D\). The distance \(A D\) is 6637.5 m . Find the deceleration of the car and the value of \(R\).

5 A particle \(P\) moves in a straight line starting from a point \(O\) and comes to rest 35 s later. At time \(t \mathrm {~s}\) after leaving \(O\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) is given by $$\begin{array} { l l } v = \frac { 4 } { 5 } t ^ { 2 } & 0 \leqslant t \leqslant 5
v = 2 t + 10 & 5 \leqslant t \leqslant 15
v = a + b t ^ { 2 } & 15 \leqslant t \leqslant 35 \end{array}$$ where \(a\) and \(b\) are constants such that \(a > 0\) and \(b < 0\).

1. Show that the values of \(a\) and \(b\) are 49 and - 0.04 respectively.
2. Sketch the velocity-time graph.
   \includegraphics[max width=\textwidth, alt={}, center]{f3ddfde8-b678-4c3a-a9a6-d984c2c38bd9-09_689_1323_349_452}
3. Find the total distance travelled by \(P\) during the 35 s .
   \includegraphics[max width=\textwidth, alt={}, center]{f3ddfde8-b678-4c3a-a9a6-d984c2c38bd9-10_487_506_260_817} Two particles of masses 1.2 kg and 0.8 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The particles hang vertically. The system is released from rest with both particles 0.64 m above the floor (see diagram). In the subsequent motion the 0.8 kg particle does not reach the pulley.
4. Show that the acceleration of the particles is \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the tension in the string.
5. Find the total distance travelled by the 0.8 kg particle during the first second after the particles are released.