MTH 425 Lecture Notes - Lecture 6: Alatau, Tencent, Exponential Growth

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15 Apr 2023

School

Ryerson University

Department

Mathematics

Course

MTH 425

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Related Questions

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?

Escape Velocity The minimum velocity required for an object to escape Earth’s gravitational pull is obtained from the solution of the equation

where v is the velocity of the object projected from Earth, y is the distance from the center of Earth, G is the gravitational constant, and M is the mass of Earth. Show that v and v are related by the equation

where v₀ is the initial velocity of the object and K is the radius of Earth.

Velocity in a Resisting Medium The velocity v of an object falling through a resisting medium such as air or water is given by

where is the initial velocity, t is the time in seconds, and k is the resistance constant of the medium. Use L’Hôpital’s Rule to find the formula for the velocity of a falling body in a vacuum by fixing and t and letting k approach zero. (Assume that the downward direction is positive.)

MTH 425 Lecture Notes - Lecture 6: Alatau, Tencent, Exponential Growth

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