Question

In the real world, falling objects (parachutists, raindrops,birds of prey, etc.) experience, in addition to the pull ofgravity, a drag force proportional to the square of the velocity.If h(t) is the height of the falling object, then dh/dt is itsvelocity, and d^2h/dt^2 is its acceleration.
Newton's law gives

d^2h/dt^2 = -g+k (dh/dt)^2

where g is the acceleration due to gravity, and k is aconstant that depends on the size and shape of the object (notethat g and k are positive). For this reason, the velocity of afalling object does not become arbitrarily large but approaches aterminal velocity. Terminal velocity is defined as the velocity atwhich acceleration is zero.

(a) Using (1), find an expression for the terminal velocity interms of g and k.
Note that, with our choice of coordinates, a falling object willhave negative velocity.

(b) A peregrine falcon is a bird of prey that hunts by tuckingin its wings and dive-bombing its prey. If the peregrine tucks inits wings, the constant k is 0.00121 m^-1 if height is measured inmeters and time in seconds. The constant g is 9.8 m/sec^2. Aperegrine falcon hunting a rabbit reaches terminal velocity 90meters above the ground.At that same moment, the rabbit sees theperegrine falcon hurtling toward it. How longdoes it have toescape?