**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)...

See similar textbooks

Similar questions

FIGURE 1: Wind tunnel experiment to measure how the force of air resistance depends on velocity. A free-falling object such as a bungee jumper is subject to the upward force of air resistance. We assumed that this force was proportional to some power b of velocity as in Fu=CaVb where Fy = the upward force of air resistance [N = kg m/s? ], Ca = drag coefficient (kg/m), and V = velocity [m/s]. An individual is suspended in a wind tunnel and the force measured for various levels of wind velocity. The result might be as listed in Table 1 V (m/s) 10 20 30 40 50 60 Fu (N) 37.9 107.3 197.2 303.6 424.3 557.7 Table 1: Experimental data for force (N) and velocity (m/s) from a wind tunnel experiment. Find least sqaures approximation Fu=CaVb to the data given in Table 1.
The acceleration of a particle moving along the = -7 t + 4 t* + 3 t2, 7. x-axis is given by the function where a is in meters per second squared and t in seconds. What is the particle's velocity at t = 2 s if the object's mass is 11.3 kg and has an initial velocity of 1.6 m/s?
A 1250 kg boat is traveling at 90 km/h when its engine is shut off. The magnitude of the frictional force fk between boat and water is proportional to the speed v of the boat. Thus, fk = 80v, where v is in meters per second and fk (the magnitude of the frictional force) is in newtons. Find the time required for the boat to slow down to 45 km/h.
You are given two masses, m₁ and m2, strapped on a pulley with a rope. m₁-13 kg and m2-17 kg. What is the equation to be used to find the system's acceleration? O a (m₂g + m₁g)/(m₂ - m₁) = a= = (m2g - m1g)/(m₂ - m₁) a = (m₂g + mig)/(m2 + m₁) O a (m₂g-m₁g)/(m₂ + m₁) =
A box lies on a frictionless table. A force of 19.5 Newtons pulls the box to the right, while a force of 20.0 N pulls the box to the left. If the acceleration of the box is –0.12 (negative values are to the left), what is the mass of the box?
An object of mass 3.1 kg with a certain initial speed on a horizontal surface comes to rest after traveling a distance of 15.9 m. If the coefficient of kinetic friction between the object and the horizontal surface is 0.27, what is the initial speed of the object? Express your answer in m/s, to at least one digit after the decimal point. plz use q =9.8
The drag force, Fa, imposed by the surrounding air on a vehicle moving with velocity Vis given by Fa Ca ApV²/2 where Ca is a constant called the drag coefficient, A is the projected frontal area of the vehicle, and p is the air density. An automobile is moving at V = 40 miles per hour with C = 0.28, A = 24 ft², and p = 0.075 lb/ft³. Determine the force, in lbf, and the power, in hp, required to overcome aerodynamic drag.
A skater with an initial speed of 8.10 m/s stops propelling himself and begins to coast across the ice, eventually coming to rest. Air resistance is negligible.(a) The coefficient of kinetic friction between the ice and the skate blades is 0.135. Find the deceleration caused by kinetic friction.(b) How far will the skater travel before coming to rest? a = + - units x = units
A block having a mass of m = 13 kg is suspended via two cables as shown in the figure. The angles shown in the figure are as follows: α = 14° and β = 32°. We will label the tension in Cable 1 as T1 and the tension in Cable 2 as T2. a) Write an expression for the sum of forces in the x direction in terms of T1, T2, m, g, α, and β. Use the specified coordinate system. b) Write an expression for the sum of forces in the y direction in terms of T1, T2, m, g, α, and β. Use the specified coordinate system.
A ball with mass 0.75 kg is thrown upward with initial velocity 25 m/s from the roof of a building 20 m high. Assume there is a force due to |v| air resistance of magnitude directed opposite to the velocity, where 30 the velocity v is measured in m/s. NOTE: Use g=9.8 m/s² as the acceleration due to gravity. Round your answers to 2 decimal places. a) Find the maximum height above the ground that the ball reaches. Height: 51.89 m Time: 5.81 X b) Find the time that the ball hits the ground. seconds X c) Use a graphing utility to plot the graphs of velocity and position versus time. Edit
The muzzle velocity of a typical 500 g spud is 25 m/s . The force given by the spud is K/(x+.09) where x is the distance of the barrel in meters. If the length of the barrel is 75 cm, what is the constant K? What is the force on the spud?
Determine the force Q-> when the block moves with constant velocity. Express your answer in vector form.