Retarding Forces

  • Sometimes the frictional force is a function of an object's velocity (such as air resistance)

  • These forces are called drag, or retarding, forces.

The Skydiver

  • Assume we drop Alex from an airplane

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Velocity as a Function of Time


Acceleration as a Function of Time

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Graph of Acceleration, Velocity, and Displacement


2005 Free Response Question 1

A ball of mass M is thrown vertically upward with an initial speed of vo. It experiences a force of air resistance given by F = —kv , where k is a positive constant. The positive direction for all vector quantities is upward. Express all algebraic answers in terms of M, k, vo, and fundamental constants. (a) Does the magnitude of the acceleration of the ball increase, decrease, or remain the same as the ball moves upward? increases Justify your answer. decreases remains the same (b) Write, but do NOT solve, a differential equation for the instantaneous speed v of the ball in terms of time t as the ball moves upward. (c) Determine the terminal speed of the ball as it moves downward. (d) Does it take longer for the ball to rise to its maximum height or to fall from its maximum height back to the height from which it was thrown? longer to rise Justify your answer. longer to fall (e) On the axes below, sketch a graph of velocity versus time for the upward and downward parts of the ball's flight, where t f is the time at which the ball returns to the height from which it was thrown.

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2013 Free Response Question 2

Mech 2. A box of mass m initially at rest is acted upon by a constant applied force of magnitude FA , as shown in the figure above. The friction between the box and the horizontal surface can be assumed to be negligible, but the box is subject to a drag force of magnitude kv where v is the speed of the box and k is a positive constant. Express all your answers in terms of the given quantities and fundamental constants, as appropriate. (a) The dot below represents the box. Draw and label the forces (not components) that act on the box.

(b) Write, but do not solve, a differential equation that could be used to determine the speed v of the box as a function of time t. If you need to draw anything other than what you have shown in part (a) to assist in your solution, use the space below. Do NOT add anything to the figure in part (a). (c) Determine the magnitude of the terminal velocity of the box. (d) Use the differential equation from part (b) to derive the equation for the speed v of the box as a function of time t. Assume that D = 0 at time t = O. (e) On the axes below, sketch a graph of the speed v of the box as a function of time t. Explicitly label any intercepts, asymptotes, maxima, or minima with numerical values or algebraic expressions, as appropriate.


F-Ev_ kvz

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