CAIE M1 (Mechanics 1) 2022 June

A car starts from rest and moves in a straight line with constant acceleration for a distance of 200 m, reaching a speed of 25 m s\(^{-1}\). The car then travels at this speed for 400 m, before decelerating uniformly to rest over a period of 5 s.

1. Find the time for which the car is accelerating. [2]
2. Sketch the velocity–time graph for the motion of the car, showing the key points. [2]
3. Find the average speed of the car during its motion. [2]

Two particles \(P\) and \(Q\), of masses 0.5 kg and 0.3 kg respectively, are connected by a light inextensible string. The string is taut and \(P\) is vertically above \(Q\). A force of magnitude 10 N is applied to \(P\) vertically upwards. Find the acceleration of the particles and the tension in the string connecting them. [5]

A crate of mass 300 kg is at rest on rough horizontal ground. The coefficient of friction between the crate and the ground is 0.5. A force of magnitude \(X\) N, acting at an angle \(\alpha\) above the horizontal, is applied to the crate, where \(\sin \alpha = 0.28\). Find the greatest value of \(X\) for which the crate remains at rest. [5]

\includegraphics{figure_4} Three coplanar forces of magnitudes 20 N, 100 N and \(F\) N act at a point. The directions of these forces are shown in the diagram. Given that the three forces are in equilibrium, find \(F\) and \(\alpha\). [6]

Two racing cars \(A\) and \(B\) are at rest alongside each other at a point \(O\) on a straight horizontal test track. The mass of \(A\) is 1200 kg. The engine of \(A\) produces a constant driving force of 4500 N. When \(A\) arrives at a point \(P\) its speed is 25 m s\(^{-1}\). The distance \(OP\) is \(d\) m. The work done against the resistance force experienced by \(A\) between \(O\) and \(P\) is 75 000 J.

1. Show that \(d = 100\). [3]

Car \(B\) starts off at the same instant as car \(A\). The two cars arrive at \(P\) simultaneously and with the same speed. The engine of \(B\) produces a driving force of 3200 N and the car experiences a constant resistance to motion of 1200 N.

1. Find the mass of \(B\). [3]
2. Find the steady speed which \(B\) can maintain when its engine is working at the same rate as it is at \(P\). [3]

A particle starts from a point \(O\) and moves in a straight line. The velocity \(v\) m s\(^{-1}\) of the particle at time \(t\) s after leaving \(O\) is given by $$v = k(3t^2 - 2t^3),$$ where \(k\) is a constant.

1. Verify that the particle returns to \(O\) when \(t = 2\). [4]
2. It is given that the acceleration of the particle is \(-13.5\) m s\(^{-2}\) for the positive value of \(t\) at which \(v = 0\). Find \(k\) and hence find the total distance travelled in the first two seconds of motion. [6]

Two particles \(A\) and \(B\), of masses 0.4 kg and 0.2 kg respectively, are moving down the same line of greatest slope of a smooth plane. The plane is inclined at 30° to the horizontal, and \(A\) is higher up the plane than \(B\). When the particles collide, the speeds of \(A\) and \(B\) are 3 m s\(^{-1}\) and 2 m s\(^{-1}\) respectively. In the collision between the particles, the speed of \(A\) is reduced to 2.5 m s\(^{-1}\).

1. Find the speed of \(B\) immediately after the collision. [2]

After the collision, when \(B\) has moved 1.6 m down the plane from the point of collision, it hits a barrier and returns back up the same line of greatest slope. \(B\) hits the barrier 0.4 s after the collision, and when it hits the barrier, its speed is reduced by 90%. The two particles collide again 0.44 s after their previous collision, and they then coalesce on impact.

1. Show that the speed of \(B\) immediately after it hits the barrier is 0.5 m s\(^{-1}\). Hence find the speed of the combined particle immediately after the second collision between \(A\) and \(B\). [7]