9709/Oct Nov/2003/3/Q9

Compressed air is escaping from a container. The pressure of the air in the container at time t is P, and the constant atmospheric pressure of the air outside the container is A. The rate of decrease of P is proportional to the square root of the pressure difference (P − A). Thus the differential equation connecting P and t is

where k is a positive constant.
(i) Find, in any form, the general solution of this differential equation.
(ii) Given that P = 5A when t = 0, and that P = 2A when t = 2, show that k =√A.
(iii) Find the value of t when P = A.
(iv) Obtain an expression for P in terms of A and t.

Solution:

Reference: PYQ - Oct/Nov 2003 Paper 3 Q9

9709/May Jun/2003/4/Q5

S₁ and S₂ are light inextensible strings, and A and B are particles each of mass 0.2 kg. Particle A is suspended from a fixed point O by the string S₁, and particle B is suspended from A by the string S₂.

The particles hang in equilibrium as shown in the diagram.

(i) Find the tensions in S₁ and S₂.

The string S₁ is cut and the particles fall. The air resistance acting on A is 0.4N and the air resistance acting on B is 0.2N.

(ii) Find the acceleration of the particles and the tension in S₂.

Solution:

Reference: PYQ - May/June 2003 Paper 4 Q5