CAIE M1 (Mechanics 1) 2017 June

1 A man pushes a wheelbarrow of mass 25 kg along a horizontal road with a constant force of magnitude 35 N at an angle of \(20 ^ { \circ }\) below the horizontal. There is a constant resistance to motion of 15 N . The wheelbarrow moves a distance of 12 m from rest.

1. Find the work done by the man.
2. Find the speed attained by the wheelbarrow after 12 m .

2
\includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-03_522_604_262_769} The four coplanar forces shown in the diagram are in equilibrium. Find the values of \(P\) and \(\theta\).

3 A train travels between two stations, \(A\) and \(B\). The train starts from rest at \(A\) and accelerates at a constant rate for \(T\) s until it reaches a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then travels at this constant speed before decelerating at a constant rate, coming to rest at \(B\). The magnitude of the train's deceleration is twice the magnitude of its acceleration. The total time taken for the journey is 180 s .

1. Sketch the velocity-time graph for the train's journey from \(A\) to \(B\).
   \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-04_496_857_516_685}
2. Find an expression, in terms of \(T\), for the length of time for which the train is travelling with constant speed.
3. The distance from \(A\) to \(B\) is 3300 m . Find how far the train travels while it is decelerating.

4 A particle \(P\) moves in a straight line starting from a point \(O\). At time \(t \mathrm {~s}\) after leaving \(O\), the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is given by \(v = ( 2 t - 5 ) ^ { 3 }\).

1. Find the values of \(t\) when the acceleration of \(P\) is \(54 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find an expression for the displacement of \(P\) from \(O\) at time \(t \mathrm {~s}\).

5 A particle is projected vertically upwards from a point \(O\) with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Two seconds later a second particle is projected vertically upwards from \(O\) with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after the second particle is projected, the two particles collide.

1. Find \(t\).
   \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-06_65_1569_488_328}
2. Hence find the height above \(O\) at which the particles collide.

6 A car of mass 1200 kg is travelling along a horizontal road.

1. It is given that there is a constant resistance to motion.
   (a) The engine of the car is working at 16 kW while the car is travelling at a constant speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the resistance to motion.
   (b) The power is now increased to 22.5 kW . Find the acceleration of the car at the instant it is travelling at a speed of \(45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. It is given instead that the resistance to motion of the car is \(( 590 + 2 v ) \mathrm { N }\) when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car travels at a constant speed with the engine working at 16 kW . Find this speed.

7
\includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-10_336_803_258_671} Two particles \(A\) and \(B\) of masses \(m \mathrm {~kg}\) and 4 kg respectively are connected by a light inextensible string that passes over a fixed smooth pulley. Particle \(A\) is on a rough fixed slope which is at an angle of \(30 ^ { \circ }\) to the horizontal ground. Particle \(B\) hangs vertically below the pulley and is 0.5 m above the ground (see diagram). The coefficient of friction between the slope and particle \(A\) is 0.2 .

1. In the case where the system is in equilibrium with particle \(A\) on the point of moving directly up the slope, show that \(m = 5.94\), correct to 3 significant figures.
2. In the case where \(m = 3\), the system is released from rest with the string taut. Find the total distance travelled by \(A\) before coming to instantaneous rest. You may assume that \(A\) does not reach the pulley.