AQA A-Level Maths: Mechanics

A particle $A$A of mass $2\ \text{kg}$2 kg lies on a rough plane inclined at $30^\circ$30∘ to the horizontal. The coefficient of friction between $A$A and the plane is $0.25$0.25. A light inextensible string is attached to $A$A, runs up a line of greatest slope and over a smooth pulley fixed at the top of the plane, and carries a particle $B$B of mass $6\ \text{kg}$6 kg hanging freely below the pulley. The system is released from rest with the string taut, and $B$B descends while $A$A moves up the plane.

(a) Draw a clear force diagram for each particle and write down the equation of motion for $A$A along the plane and for $B$B. (4 marks)

(b) Find the acceleration of the system and the tension in the string. (5 marks)

(c) When $B$B has descended $1\ \text{m}$1 m it strikes the ground and the string immediately becomes slack. Assuming $A$A does not reach the pulley, find how much further up the plane $A$A travels before it first comes to instantaneous rest. (3 marks)

(Take $g = 9.8\ \text{m s}^{-2}$g=9.8 m s−2.)

A ball is kicked from a point $O$O on horizontal ground with speed $28\ \text{m s}^{-1}$28 m s−1 at an angle of $30^\circ$30∘ above the horizontal. The ball is modelled as a particle moving freely under gravity, and it lands back on the same horizontal ground.

(a) Find the time taken for the ball to reach its greatest height, and the greatest height reached. (4 marks)

(b) Find the total time the ball is in the air, and hence the horizontal distance from $O$O to the point where it lands. (3 marks)

(c) Find the speed of the ball and the direction in which it is moving $1\ \text{second}$1 second after it is kicked. (3 marks)

(Take $g = 9.8\ \text{m s}^{-2}$g=9.8 m s−2.)

A particle $P$P moves along a straight line. At time $t$t seconds, where $t \geq 0$t≥0, the velocity of $P$P is

$v = 3t^2 - 24t + 36 \quad \text{m s}^{-1}.$v=3t2−24t+36m s−1.

When $t = 0$t=0 the particle is at the origin $O$O.

(a) Find the values of $t$t at which $P$P is instantaneously at rest, and the acceleration of $P$P at each of these instants. (4 marks)

(b) Find the total distance travelled by $P$P during the first $4$4 seconds of its motion. (4 marks)

A uniform plank $AB$AB has length $5\ \text{m}$5 m and mass $30\ \text{kg}$30 kg. It rests horizontally in equilibrium on two smooth supports at the points $C$C and $D$D, where $AC = 1\ \text{m}$AC=1 m and $AD = 3.5\ \text{m}$AD=3.5 m. A child of mass $25\ \text{kg}$25 kg, modelled as a particle, stands on the plank and walks slowly from $A$A towards $B$B.

Find how far beyond $D$D, towards $B$B, the child can stand before the plank is about to tilt, giving the distance of this point from $A$A.

(6 marks)

Two small smooth spheres $A$A and $B$B of equal radius move towards each other along the same straight horizontal line. Sphere $A$A has mass $3\ \text{kg}$3 kg and is moving with speed $5\ \text{m s}^{-1}$5 m s−1; sphere $B$B has mass $2\ \text{kg}$2 kg and is at rest. The spheres collide directly. Immediately after the collision $A$A continues to move in its original direction with speed $1\ \text{m s}^{-1}$1 m s−1.

(a) Find the speed of $B$B immediately after the collision. (3 marks)

(b) Find the magnitude of the impulse exerted on $B$B in the collision. (2 marks)

A crate of mass $5\ \text{kg}$5 kg is pulled across rough horizontal ground by a light rope. The rope is held taut at $30^\circ$30∘ above the horizontal and the tension in it is $20\ \text{N}$20 N. The coefficient of friction between the crate and the ground is $0.2$0.2. The crate is modelled as a particle and remains in contact with the ground.

Find the acceleration of the crate.

(4 marks)