WME01 Mechanics 1 (M1): January 2026 Paper 01

A particle P of mass 3m and a particle Q of mass 5m are on a smooth horizontal surface. The particles move towards each other in opposite directions along the same straight line and collide.

Immediately before the collision, the speed of P is 3u and the speed of Q is 4u

The magnitude of the impulse exerted on P by Q in the collision is mu

(i)the speed of P immediately after the collision,

(ii)the speed of Q immediately after the collision.[6]

speed (m s^–1)

A particle P starts from rest at time t = 0 and moves in a straight line

• from t = 0 to t = 12 s, P accelerates uniformly until it reaches a speed of 3U m s^–1

• from t = 12 s to t = 31 s, P moves with constant speed 3U m s^–1

• from t = 31 s to t = 41 s, P decelerates uniformly until it has speed 2U m s^–1

as shown on the speed‑time graph in Figure 1.

The distance travelled by P between t = 0 and t = 41 s is 600 m.

(a)Show that U = 6[3]

(b)Find the distance travelled by P whilst moving from rest to a speed of 6 m s^–1[2]

(c)Find the acceleration of P between t = 0 and t = 12 s.[1]

(d)Find the deceleration of P between t = 31 s and t = 41 s.[2]

(e)Sketch an acceleration‑time graph to represent the motion of P from t = 0 to t = 41 s.[2]

A particle moves on a smooth horizontal plane under the action of three horizontal forces, F₁, F₂ and F₃, where

where p and q are constants.

The resultant force acts in the direction of the vector (4i – 5j)

(a)Show that

[4]

The mass of the particle is 0.5 kg.

Given that p = 7

(b)find the acceleration of the particle.[4]

Figure 2 shows a uniform rod AB of length 6 m suspended by two light vertical ropes.

The first rope is attached to the rod at A. The second rope is attached to the rod at the point C, where CB = 0.4 m.

The rod has mass 30 kg.

A particle of mass 20 kg is now attached to the rod at B.

The rod is in equilibrium in a horizontal position.

(a)Find

(i)the tension in the rope attached to the rod at C,

(ii)the tension in the rope attached to the rod at A.[6]

The particle of mass 20 kg at B is removed and replaced by a particle of mass M kg.

The rod remains in equilibrium in a horizontal position.

(b)Find the exact maximum value of M.[3]

The points O, A, B and C are on a rough horizontal surface where OABC is a straight line.

A particle is projected horizontally from O and slides across the surface, passing through A, B and C in that order.

The particle passes through

• A with speed U m s^–1

• B with speed U2 m s^–1

The coefficient of friction between the particle and the surface is 17

It takes 0.75 s for the particle to move from A to B.

(a)Show that U = 2.1 to 2 significant figures.[6]

The particle passes through C with speed m s

(b)Find the time taken to move from B to C.[3]

(c)Find the distance AC.[3]

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

At 9 am, a ship S leaves a harbour and moves with constant velocity (10i + 15j) km h^–1

(a)Find the exact speed of S[2]

The harbour is at the origin O.

At 9 am, a boat B leaves the point with position vector (50i + 10j) km and moves with constant velocity (–8i + 10j) km h^–1

(b)Find the direction in which B is moving, giving your answer as a bearing to the nearest degree.[3]

(c)Show that at time t hours after 9 am,

[4]

(d)Show that B and S do not collide.[3]

When B is north‑east of S, the distance between S and B is d km.

(e)Find the value of d.[4]

Bucket (0.45 kg)

A rough plane is inclined at an angle θ to the horizontal, where tan θ = 43

One end of a light inextensible string is attached to a package P. The string passes over a smooth pulley that is fixed at the top of the plane. The string from P to the pulley lies along a line of greatest slope of the plane. The other end of the string is attached to a bucket. The package P has mass 1 kg and is held at rest on the plane. The bucket has mass 0.45 kg and hangs vertically below the pulley with the string taut, as shown in Figure 3.

The coefficient of friction between P and the plane is μ

When a small block of mass 0.2 kg is placed in the bucket, the system is released and remains at rest in limiting equilibrium, with P on the point of slipping down the plane.

(a)Show that μ =

[7]

Additional identical blocks of mass 0.2 kg are added one at a time into the bucket until P starts to move up the plane. At the instant when P starts to move up the plane, there are a total of n blocks in the bucket.

(b)Find

(i)the value of n

(ii)the magnitude of the initial acceleration of P[7]