CAIE M1 (Mechanics 1) 2023 November

A block of mass 15 kg slides down a line of greatest slope of an inclined plane. The top of the plane is at a vertical height of 1.6 m above the level of the bottom of the plane. The speed of the block at the top of the plane is 2 m s\(^{-1}\) and the speed of the block at the bottom of the plane is 4 m s\(^{-1}\). Find the work done against the resistance to motion of the block. [4]

\includegraphics{figure_2} The diagram shows a smooth ring \(R\), of mass \(m\) kg, threaded on a light inextensible string. A horizontal force of magnitude 2 N acts on \(R\). The ends of the string are attached to fixed points \(A\) and \(B\) on a vertical wall. The part \(AR\) of the string makes an angle of 30° with the vertical, the part \(BR\) makes an angle of 40° with the vertical and the string is taut. The ring is in equilibrium. Find the tension in the string and find the value of \(m\). [5]

\includegraphics{figure_3} A block of mass 10 kg is at rest on a rough plane inclined at an angle of 30° to the horizontal. A force of 120 N is applied to the block at an angle of 20° above a line of greatest slope (see diagram). There is a force resisting the motion of the block and 200 J of work is done against this force when the block has moved a distance of 5 m up the plane from rest. Find the speed of the block when it has moved a distance of 5 m up the plane from rest. [5]

A particle \(P\) of mass 0.2 kg lies at rest on a rough horizontal plane. A horizontal force of 1.2 N is applied to \(P\).

1. Given that \(P\) is in limiting equilibrium, find the coefficient of friction between \(P\) and the plane. [3]
2. Given instead that the coefficient of friction between \(P\) and the plane is 0.3, find the distance travelled by \(P\) in the third second of its motion. [4]

A particle \(A\) of mass 0.5 kg is projected vertically upwards from horizontal ground with speed 25 m s\(^{-1}\).

1. Find the speed of \(A\) when it reaches a height of 20 m above the ground. [2]

When \(A\) reaches a height of 20 m, it collides with a particle \(B\) of mass 0.3 kg which is moving downwards in the same vertical line as \(A\) with speed 32.5 m s\(^{-1}\). In the collision between the two particles, \(B\) is brought to instantaneous rest.

1. Show that the velocity of \(A\) immediately after the collision is 4.5 m s\(^{-1}\) downwards. [2]
2. Find the time interval between \(A\) and \(B\) reaching the ground. You should assume that \(A\) does not bounce when it reaches the ground. [4]

A railway engine of mass 120000 kg is towing a coach of mass 60000 kg up a straight track inclined at an angle of \(\alpha\) to the horizontal where \(\sin \alpha = 0.02\). There is a light rigid coupling, parallel to the track, connecting the engine and coach. The driving force produced by the engine is 125000 N and there are constant resistances to motion of 22000 N on the engine and 13000 N on the coach.

1. Find the acceleration of the engine and find the tension in the coupling. [5]

At an instant when the engine is travelling at 30 m s\(^{-1}\), it comes to a section of track inclined upwards at an angle \(\beta\) to the horizontal. The power produced by the engine is now 4500000 W and, as a result, the engine maintains a constant speed.

1. Assuming that the resistance forces remain unchanged, find the value of \(\beta\). [4]

A particle \(X\) travels in a straight line. The velocity of \(X\) at time \(t\) s after leaving a fixed point \(O\) is denoted by \(v\) m s\(^{-1}\), where $$v = -0.1t^3 + 1.8t^2 - 6t + 5.6.$$ The acceleration of \(X\) is zero at \(t = p\) and \(t = q\), where \(p < q\).

1. Find the value of \(p\) and the value of \(q\). [4]

It is given that the velocity of \(X\) is zero at \(t = 14\).

1. Find the velocities of \(X\) at \(t = p\) and at \(t = q\), and hence sketch the velocity-time graph for the motion of \(X\) for \(0 \leq t \leq 15\). [3]
2. Find the total distance travelled by \(X\) between \(t = 0\) and \(t = 15\). [5]