This question models the motion of a person diving from a 10-metre board into a swimming pool. The whole motion may be split into two phases, above and below the water.
In the first phase the diver, who can be considered as a particle of mass 60, jumps vertically upwards off the diving board with initial speed 5. The diver misses the board on return, before entering the water 10m below the level of the diving board. All frictional forces (including air resistance) may be ignored.
(a) Consider the first phase of the motion, while the diver is still in the air.
(i) Draw a force diagram showing all forces acting on the diver. Choose an appropriate coordinate system and axes. Express the initial (vector) displacement and the initial velocity in terms of your chosen coordinate system. Determine the equation of motion of the diver, and state the initial condition.
(ii) Find the maximum height above the board that the diver attains in terms of g, the magnitude of the acceleration due to gravity, and the speed at which the diver enters the swimming pool, in terms of g.
The second phase of the motion is underwater. Here the diver experiences a force of resistance, due to the water opposing the motion, of -18gv, where v is the velocity of the diver in ms-1, and g is the magnitude of the acceleration due to gravity. In addition to the force of gravity, the diver also experiences a constant upward buoyancy force of 72gN.
(b) For this phase, while the diver is travelling downwards in the water, take the origin O of the coordinate system to be the point at which the diver entered the water. Take the unit vector i pointing vertically downwards.
(i) Draw a force diagram showing all the forces acting on the diver. Clearly identify each force.
(ii) Choose and define an appropriate coordinate system and axes. Express the initial (vector) displacement and the initial velocity in terms of your chosen coordinate system. Express each force in terms of the unit vectors that you have defined.
(iii) Hence show that the equation of motion can be expressed as
v * dv/dx =( -g/10 ) * (2+3v)
where v is the speed of the diver at a distance x below the surface of the water.
(iv) Using the substitution u = 2+ 3v, or otherwise, find the particular solution of the differential equation, giving x in terms of v.
(v) Determine how deep the diver will submerge, giving your answer correct to one decimal place, using g = 9.81.