WME01 Mechanics 1 (M1): May/June 2026 Paper 01

Railway truck A has mass 5m and railway truck B has mass 3m. The trucks are moving towards each other in opposite directions along the same straight track when they collide directly. Immediately before the collision the speed of A is u and the speed of B is 1.5u

As a result of the collision, the trucks couple together.

(a)Find, in terms of u, the common speed of the trucks immediately after the collision.[3]

(b)Find, in terms of m and u, the magnitude of the impulse exerted on B by A in the collision.[3]

A non‑uniform rod AB has mass 20 kg and length 6 m. The rod is held in equilibrium in a horizontal position by two light vertical strings. One string is attached to the rod at P, where AP = 0.5 m and the other is attached to the rod at Q, where QB = 1.5 m, as shown in Figure 1.

The centre of mass of the rod is 4 m from A.

(a)Find the tension in the string attached to the rod at Q.[3]

A particle is now attached to the rod at B.

The rod remains in equilibrium in a horizontal position but is on the point of tilting.

The tension in the string attached to the rod at Q is now X newtons.

(b)Find the value of X.[3]

A small ball of mass 0.04 kg moves along a straight line on a pinball table.

At time t = 0, the ball passes through the point O

The displacement of the ball is measured from O

Figure 2 shows the displacement‑time graph for the motion of the ball from t = 0 to t = 0.5

(a)State the displacement of the ball from O at t = 0.5[1]

(b)Find the total distance moved by the ball between t = 0 and t = 0.5[1]

(c)Sketch a velocity‑time graph for the motion of the ball from t = 0 to t = 0.5[4]

(d)Find the magnitude of the impulse exerted on the ball at t = 0.3, stating the units of your answer.[4]

The point O is 28.4 m above a horizontal plane. A small ball is projected vertically upwards from O with speed U m s^–1 and first hits the plane with speed 24 m s^–1

Air resistance is assumed to be negligible.

(a)Find the value of U.[3]

Each time the ball hits the plane, the speed of the ball is instantaneously halved.

(b)Find the total time, after the ball has hit the plane for the first time, for which the ball is at least 4 m above the plane.[6]

A particle P, whose weight is 50 N, is at rest on a rough plane. The plane is inclined to the horizontal at 20° and the coefficient of friction between P and the plane is 0.3

The particle P is held in equilibrium by a force of magnitude X newtons.

The line of action of the force makes an angle of 40° with the plane, as shown in Figure 3. The line of action of the force lies in a vertical plane which contains a line of greatest slope of the plane.

Given that P is on the point of sliding down the plane, find the value of X.[6]

The resultant R of two forces F₁ N and F₂ N is (–5i + 6j) N.

(a)Find the magnitude of R.[2]

(b)Find, in degrees, the angle between the direction of R and the direction of the vector (2i + j). Show your method clearly.[3]

Given that F₁ is parallel to the vector (–i + 2j) and that F₂ is parallel to the vector (2i – 3j),

(c)find F₁ and F₂[4]

[ In this question i and j are horizontal unit vectors directed due east and due north

A boat, which is modelled as a particle, is moving with constant velocity (–2i + j) km h^–1

At time t = 1.5 hours after noon, the boat passes through the point with position vector (11i + 3.5j) km.

(a)Show that, at time t hours after noon, the position vector of the boat is

[3]

The boat is visible from O when it is 9 km or less from O.

(b)Find the length of time for which the boat is visible from O, showing your method clearly.[4]

(c)Find the value of t when the boat is northwest of O.[2]

At the instant when the boat is at the point P, OP is perpendicular to the direction in which the boat is moving.

(d)Show that OP = 185 km

(Solutions relying on calculator technology are not acceptable.)[6]

One end of a light inextensible string is attached to a particle A of mass 3m. The other end of the string is attached to a particle B of mass 2m. Particle A lies on a rough plane

which is inclined to the horizontal at an angle α, where tan α = 34 . Particle B lies on a

smooth horizontal plane. The string passes over a small smooth pulley P which is fixed at the top of the inclined plane, as shown in Figure 4.

The two particles, the string and the pulley all lie in the same vertical plane. The portion of the string between A and P lies along a line of greatest slope of the inclined plane.

The coefficient of friction between A and the inclined plane is

The system is released from rest with the string taut and A begins to move down the inclined plane.

(a)Find, in terms of m and g, the magnitude of the frictional force acting on A.[3]

(b)Write down an equation of motion for B.[1]

(c)Show that the tension in the string is

[4]

(d)State how you have used the fact that the string is inextensible.[1]

The force exerted on the pulley by the string is S.

(e)State clearly, giving your answer in terms of α, the direction of S.[1]

(f)Find, in terms of m and g, the magnitude of S.[4]