WME01 Mechanics 1 (M1): January 2021 Paper 01

A small stone is projected vertically upwards with speed 20 m s^–1 from a point O which is 5 m above horizontal ground. The stone is modelled as a particle moving freely under gravity.

Find

(a)the speed of the stone at the instant when it is 2 m above the ground,[2]

(b)the total time between the instant when the stone is projected from O and the instant when it first strikes the ground.[4]

Two particles, P and Q, have masses 2m and m respectively. The particles are moving towards each other in opposite directions along the same straight line on a smooth horizontal plane. The particles collide directly.

Immediately before the collision, the speed of P is 3u and the speed of Q is 2u.

The magnitude of the impulse exerted on Q by P in the collision is 5mu.

Find

(a)the speed of P immediately after the collision,[3]

(b)the speed of Q immediately after the collision.[3]

A parcel of mass 20 kg is at rest on a rough horizontal floor. The coefficient of friction between the parcel and the floor is 0.3

Two forces, both acting in the same vertical plane, of magnitudes 200 N and T N are applied to the parcel. The line of action of the 200 N force makes an angle of 15° with the horizontal and the line of action of the T N force makes an angle of 25° with the horizontal, as shown in Figure 1. The parcel is modelled as a particle P.

Find the smallest value of T for which P remains in equilibrium.[9]

A metal girder AB has weight W newtons and length 6 m. The girder rests in a horizontal position on two supports C and D where AC = DB = 1 m, as shown in Figure 2.

When a force of magnitude 900 N is applied vertically upwards to the girder at A, the girder is about to tilt about D.

When a force of magnitude 1500 N is applied vertically upwards to the girder at B, the girder is about to tilt about C.

The girder is modelled as a non-uniform rod whose centre of mass is a distance x metres from A.

Find the value of x.[6]

A particle is acted upon by two forces F and G. The force F has magnitude 8 N and acts in a direction with a bearing of 240°. The force G has magnitude 10 N and acts due South.

Given that R = F + G, find

(i)the magnitude of R,

(ii)the direction of R, giving your answer as a bearing to the nearest degree.[7]

Two girls, Agatha and Brionie, are roller skating inside a large empty building. The girls are modelled as particles.

At time t = 0, Agatha is at the point with position vector (11i + 11j) m and Brionie is at the point with position vector (7i + 16j) m. The position vectors are given relative to the door, O, and i and j are horizontal perpendicular unit vectors.

Agatha skates with constant velocity (3i – j) m s^–1

Brionie skates with constant velocity (4i – 2j) m s^–1

(a)Find the position vector of Agatha at time t seconds.[2]

At time t = 6 seconds, Agatha passes through the point P.

(b)Show that Brionie also passes through P and find the value of t when this occurs.[4]

At time t seconds, Agatha is at the point A and Brionie is at the point B.

(c)Show that AB = [(t – 4)i + (5 – t)j] m[2]

(d)Find the distance between the two girls when they are closest together.[4]

A helicopter is hovering at rest above horizontal ground at the point H. A parachutist steps out of the helicopter and immediately falls vertically and freely under gravity from rest for 2.5 s. His parachute then opens and causes him to immediately decelerate at a constant rate of 3.9 m s^–2 for T seconds (T < 6), until his speed is reduced to V m s^–1. He then moves with this constant speed V m s^–1 until he hits the ground. While he is decelerating, he falls a distance of 73.75 m. The total time between the instant when he leaves H and the instant when he hits the ground is 20 s.

The parachutist is modelled as a particle.

(a)Find the speed of the parachutist at the instant when his parachute opens.[1]

(b)Sketch a speed-time graph for the motion of the parachutist from the instant when he leaves H to the instant when he hits the ground.[2]

(c)Find the value of T.[5]

(d)Find, to the nearest metre, the height of the point H above the ground.[4]

Two particles, A and B, have masses 2 kg and 4 kg respectively. The particles are connected by a light inextensible string. The string passes over a small smooth pulley which is fixed

at the top of a rough plane. The plane is inclined to the horizontal ground at an angle α

where tan α = 34 . The particle A is held at rest on the plane at a distance d metres from

the pulley. The particle B hangs freely at rest, vertically below the pulley, at a distance h metres above the ground, as shown in Figure 3. The part of the string between A and the pulley is parallel to a line of greatest slope of the plane. The coefficient of friction

between A and the plane is 14

The system is released from rest with the string taut and B descends.

(a)Find the tension in the string as B descends.[9]

On hitting the ground, B immediately comes to rest.

Given that A comes to rest before reaching the pulley,

(b)find, in terms of h, the range of possible values of d.[7]

(c)State one physical factor, other than air resistance, that could be taken into account to make the model described above more realistic.[1]