A vertical cylindrical water tank of diameter D=1 m is being drained via a hole of diameter d = 5 cm at the bottom of the tank. The speed of the water through the hole is 2gh, where h(t) is the depth of the water at time t and g is the acceleration due to gravity (9.81 m/s). (i) Show that the differential equation describing the drainage is: dh + k/=0 where k = 28 ( 1 ) [5 marks] dt (ii) Use this equation to calculate how long it will take to drain the tank if the initial depth of water is 1.5 m. [5 marks] A projectile is launched from flat, horizontal ground and follows a trajectory described by the position vector r(t)=[10t+5(1e')]15tj+(20t-5t)k, where t represents time and the vector k points vertically upwards. Calculate: (i) The maximum height achieved. [3 marks] (ii) The positions at which the projectile is at 75% of its maximum height. [4 marks] (iii) The speed at which the projectile hits the ground. [3 marks] The following equation of state (in which R is a constant) relates the pressure in a non-ideal gas to its volume and temperature: RT P(V,T)=; 2 V-4 V(V+4)T Write this equation in the form of its total differential, dp. [5 marks]