**Air Resistance and Free Fall Dynamics**The resistance of air to the motion of bodies in free fall depends on many factors, such as the size of the body and its shape, the density and temperature of the air, and the velocity of the body falling through the air. A useful assumption, only approximately true, is that the resisting force takes the form \(\vec{F}_R = -k \vec{v}\), where \(k\) is a constant whose value in any particular case is determined by factors other than velocity.Consider the free fall of an object released from rest.**(a)** Show that Newton’s second law gives\[ ma = mg - kv \]which is equivalent to \[ m \frac{d^2y}{dt^2} = mg - k \frac{dy}{dt} \]**(b)** What are the dimensions of \(k\), in terms of mass (M), length (L), and time (T)?**(c)** Show that the body ceases to accelerate when it reaches a velocity \( v_T = \frac{mg}{k} \) called the terminal velocity.**(d)** Prove, by substitution into the equation of part (a), that the velocity varies with time as\[ v = v_T \left( 1 - e^{-kt/m} \right) \]**(e)** Make sketches of \(v(t)\) and \(y(t)\). Describe any significant features of your sketches.

Please see the answer below. Note: The first three sub-parts are answered below by following the…

The resistance of air to the motion of bodies in free fall depends on many factors, such as the size of the body and its shape, the density and temperature of the air, and the velocity of the body falling through the air. A useful assumption, only approximately true, is that the resisting force takes the form FR = -kủ, where k is a constant whose value in any particular case is determined by factors other than velocity. Consider free fall of an object released from rest. (a) Show that Newton's second law gives d²y dy = mg – k dt та — тg — ku which is equivalent to m- dt2 (b) What are the dimensions of k, in terms of mass (M), length (L), and time (T)? (c) Show that the body ceases to accelerate when it reaches a velocity vr = mg/k called the terminal velocity. (d) Prove, by substitution into the equation of part (a), that the velocity varies with time as v = Ur (1 – e-kt/m) (e) Make sketches of v(t) and y(t). Describe any significant features of your sketches.

The resistance of air to the motion of bodies in free fall depends on many factors, such as the size of the body and its shape, the density and temperature of the air, and the velocity of the body falling through the air. A useful assumption, only approximately true, is that the resisting force takes the form FR = -kủ, where k is a constant whose value in any particular case is determined by factors other than velocity. Consider free fall of an object released from rest. (a) Show that Newton's second law gives d²y dy = mg – k dt та — тg — ku which is equivalent to m- dt2 (b) What are the dimensions of k, in terms of mass (M), length (L), and time (T)? (c) Show that the body ceases to accelerate when it reaches a velocity vr = mg/k called the terminal velocity. (d) Prove, by substitution into the equation of part (a), that the velocity varies with time as v = Ur (1 – e-kt/m) (e) Make sketches of v(t) and y(t). Describe any significant features of your sketches.

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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