**Understanding Motion with Air Resistance**When you drop an object of mass \( m \) from a tall building, the primary forces affecting its motion are gravity and air resistance. Air resistance is proportional to the object's speed with a positive constant of proportionality \( k \). Let \( g \) represent gravitational acceleration (a positive constant).**Expressing Total Force**To express the total force \( F \) in terms of \( m \), \( g \), and the object's velocity \( v \) (considering upward displacement as positive), the equation is:\[F = mg - kv\]**Newton’s Second Law**According to Newton’s second law, force equals mass times acceleration, expressed as \( F = ma \). By relating acceleration to velocity, we can rewrite the total force equation as a first-order differential equation for \( v \) as a function of time \( t \). Denote acceleration \( v' \) as \( \frac{dv}{dt} \).**Differential Equation**To solve the differential equation,\[m \left(-\frac{dv}{dt}\right) \]results in a syntax error indicating that this is not an equation as presented.**Solving for Velocity**Given the initial condition \( v(0) = v_0 \), solving the differential equation gives:\[v(t) = \frac{mg}{k} \left(1 - e^{-\frac{k}{m}t}\right)\]**Finding Terminal Velocity**The terminal velocity is determined when the object's speed stops changing:\[\text{Terminal velocity} = \frac{mg}{k}\]*Note: The boxes above indicate where certain expressions need correction.*

Answered: Image /qna-images/answer/abf29cbb-a8ae-439f-86ca-2d69bc534699.jpg

You drop an object of mass m from a tall building. Suppose the only forces affecting its motion are gravity, and air resistance proportional to the object's speed with positive constant of proportionality k. Let g denote gravitational acceleration (a positive constant). Express the total force in terms of m, g, and the object's velocity v, where upward displacement is considered positive. F - Newton's second law tells us that force is equal to mass x acceleration, F = ma. Relating mg - kv acceleration to velocity, rewrite the equation for total force above as a first order differential equation for v as a function of t. Denote v' as dv dt dv dt this is not an equation. m v(t): = Solve this differential equation for v(t) with the initial condition v(0) = V0. mg k 1 e X m Find the terminal velocity. Terminal velocity = mg k X syntax error: X X

You drop an object of mass m from a tall building. Suppose the only forces affecting its motion are gravity, and air resistance proportional to the object's speed with positive constant of proportionality k. Let g denote gravitational acceleration (a positive constant). Express the total force in terms of m, g, and the object's velocity v, where upward displacement is considered positive. F - Newton's second law tells us that force is equal to mass x acceleration, F = ma. Relating mg - kv acceleration to velocity, rewrite the equation for total force above as a first order differential equation for v as a function of t. Denote v' as dv dt dv dt this is not an equation. m v(t): = Solve this differential equation for v(t) with the initial condition v(0) = V0. mg k 1 e X m Find the terminal velocity. Terminal velocity = mg k X syntax error: X X

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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