Dynamics of a particle

Newton's laws, connected particles, friction, momentum and impulse.

Two particles $P$ and $Q$ , of masses $0.3\,\text{kg}$ and $0.2\,\text{kg}$ , move towards each other along the same straight line on a smooth horizontal surface. Just before they collide, $P$ has speed $6\,\text{m s}^{-1}$ and $Q$ has speed $2\,\text{m s}^{-1}$ . Immediately after the collision, $P$ continues in the same direction with speed $1\,\text{m s}^{-1}$ .

(a) Find the speed of $Q$ immediately after the collision.

(b) Find the magnitude of the impulse exerted on $Q$ in the collision.

(a) M1 Conservation of momentum (direction of $P$ positive): $0.3 \times 6 - 0.2 \times 2 = 0.3 \times 1 + 0.2v$ (1)

A block $A$ of mass $4\,\text{kg}$ rests on a rough horizontal table. The coefficient of friction between $A$ and the table is $0.25$ . $A$ is attached to one end of a light inextensible string which passes over a small smooth pulley at the edge of the table. The other end of the string is attached to a particle of mass , which hangs freely above the floor. The system is released from rest with the string taut. hits the floor and does not rebound; does not reach the pulley.

(a) B1 $R = 4g$ and friction $F = 0.25 \times 4g = 9.8\,\text{N}$ (1)
M1 Equations of motion, e.g. $B$ : $2g - T = 2a$ and $A$ : $T - 9.8 = 4a$ (1)
A1 $a = 1.6\,\text{m s}^{-2}$ ( $1.63$ ) (1)
A1 $T = 16\,\text{N}$ ( $16.3$ ) (1)
(b) M1 Speed when $B$ hits the floor: $v^2 = 2 \times 1.633 \times 0.5$ , $v = 1.28\,\text{m s}^{-1}$ (1)
B1 Afterwards the string is slack and $A$ decelerates at $0.25g = 2.45\,\text{m s}^{-2}$ (1)
A1 Distance $= \dfrac{1.633}{2 \times 2.45} = 0.33\,\text{m}$ (1)

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