A humanitarian mission aims to drop an aid box into an isolated disaster area. The box weighs m kg and is dropped from a helicopter, with initial zero speed. In free fall, the box is subjected to an air resistance force that is proportional to its speed, with B being the proportionality constant. Consider gravity to be g.
(a) Find the second-order differential equation that describes the movement of the box in the vertical axis. Consider the position of the box just before being dropped to be y(0) = 0.
(b) Solve this equation and obtain expressions for the vertical position and velocity of the box, in terms of B, m and g. Use the initial conditions.
(c) Consider the values m = 1000 kg, B = 500 kg/s and g = 10 m/s2 and compute the final speed of the box (steady state solution).
(d) The speed obtained in part (c) is too high and would damage the box when landing. To avoid this, the humanitarian agency has attached the box to a parachute thereby increasing the air resistance on the box-parachute system. The new value of B is 5000 kg/s. Assuming that the system is in steady state before the opening of the parachute (that is, it has reached the speed found in part (c)), find the minimum height at which the parachute needs to open such that its speed at landing is less than 3 m/s.