WME01 Mechanics 1 (M1): May/June 2023 Paper 01

A particle A has mass 4 kg and a particle B has mass 2 kg.

The particles move towards each other in opposite directions along the same straight line on a smooth horizontal table and collide directly.

Immediately before the collision, the speed of A is 2u m s^–1 and the speed of B is 3u m s^–1

Immediately after the collision, the speed of B is 2u m s^–1

The direction of motion of B is reversed by the collision.

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

(b)State the direction of motion of A immediately after the collision.[1]

(c)Find, in terms of u, the magnitude of the impulse received by B in the collision. State the units of your answer.[3]

A particle P rests in equilibrium on a smooth horizontal plane.

A system of three forces, F₁ N, F₂ N and F₃ N where

is applied to P.

Given that P remains in equilibrium,

(a)find F₃ in terms of c, i and j.[2]

The force F₃ is removed from the system.

Given that c = 2

(b)find the size of the angle between the direction of i and the direction of the resultant force acting on P.[4]

The mass of P is m kg.

Given that the magnitude of the acceleration of P is 8.5 m s^–2

(c)find the value of m.[4]

Two students observe a book of mass 0.2 kg fall vertically from rest from a shelf that is 1.5 m above the floor.

Student A suggests that the book is modelled as a particle falling freely under gravity.

(a)Use student A’s model to find the time taken for the book to reach the floor.[3]

Student B suggests an improved model where the book is modelled as a particle experiencing a constant resistance to motion of magnitude R newtons.

Given that the time taken for the book to reach the floor is 0.6 seconds,

(b)use student B’s model to find the value of R[5]

Figure 1 shows a beam AB, of mass m kg and length 2 m, suspended by two light vertical ropes. The ropes are attached to the points C and D on the beam, where AC = 0.6 m and DB = 0.2 m The beam is in equilibrium in a horizontal position.

A particle of mass pm kg is attached to the beam at A and the beam remains in equilibrium in a horizontal position.

The beam is modelled as a uniform rod.

(a)Given that the tension in the rope attached at C is four times the tension in the rope attached at D, use the model to find the exact value of p.[7]

The particle of mass pm kg at A is removed and replaced by a particle of mass qm kg at A. The beam remains in equilibrium in a horizontal position but is now on the point of tilting.

(b)Using the model, find the exact value of q[4]

(c)State how you have used the modelling assumption that the beam is uniform.[1]

speed (m s^–1)

The speed-time graph in Figure 2 illustrates the motion of a car travelling along a straight horizontal road. At time t = 0 , the car starts from rest and accelerates uniformly for 30 s until it reaches a speed of V m s^–1 The car then travels at a constant speed of V m s^–1 until time t = T seconds.

(a)Show that the distance travelled by the car between t = 0 and t = T seconds is V (T – 15) metres.[2]

A motorbike also travels along the same road.

• The motorbike starts from rest at time t = 10 s and accelerates uniformly for 40 s

• The acceleration of the motorbike is the same as the acceleration of the car

• The motorbike then travels at a constant speed for a further 10 s before decelerating uniformly until it reaches a speed of V m s^–1 at time T seconds

(b)On Figure 2, sketch a speed-time graph for the motion of the motorbike.

[If you need to redraw your sketch, there is a copy of Figure 2 on page 15.][2]

(c)Show that the constant speed of the motorbike is 4V3 m s^–1[2]

At time t = T seconds, the distance travelled by each vehicle is the same.

(d)Find the value of T[5]

A particle of weight W newtons lies at rest on a rough horizontal surface, as shown in Figure 3. A force of magnitude P newtons is applied to the particle.

The force acts at an angle α to the horizontal, where tan α = 43

The coefficient of friction between the particle and the surface is 14

Given that the particle does not move, show that

[7]

A car of mass 1200 kg is towing a trailer of mass 600 kg up a straight road, as shown in Figure 4.

The road is inclined at an angle α to the horizontal, where sin α = 112

The driving force produced by the engine of the car is 3000 N.

The car moves with acceleration 0.75 m s^–2

The non-gravitational resistance to motion of

• the car is modelled as a constant force of magnitude 2R newtons

• the trailer is modelled as a constant force of magnitude R newtons

The car and the trailer are modelled as particles. The tow bar between the car and trailer is modelled as a light rod that is parallel to the direction of motion.

Using the model,

(a)show that the value of R is 60[4]

(b)find the tension in the tow bar.[3]

When the car and trailer are moving at a speed of 12 m s^–1 , the tow bar breaks.

Given that the non-gravitational resistance to motion of the trailer remains unchanged,

(c)use the model to find the further distance moved by the trailer before it first comes to rest.[4]

A square floor space ABCD, with centre O, is modelled as a flat horizontal surface measuring 50 m by 50 m, as shown in Figure 5. → → The horizontal unit vectors i and j are in the direction of AB and AD respectively.

All position vectors are given relative to O.

A small robot R is programmed to travel across the floor at a constant velocity.

• At time t = 0 , R is at the point with position vector (–2i + j) m

• At time t = 11 s , R is at the point with position vector (9i + 23j) m

• At time t seconds, the position vector of R is r metres

(a)Find, in terms of t, i and j, an expression for r[3]

A second robot S is at the point C.

• At time t = 0 , S leaves C and moves with constant velocity (–i – j) m s^–1

• At time t seconds, the position vector of S is s metres

(b)Write down, in terms of t, i and j, an expression for s[1]

(c)Show that

[2]

(d)Find the time when the distance between R and S is a minimum.[3]