Problem Like friction, drag force opposes the motion of a particle in a fluid; however, drag force depends on the particle's velocity. Find the expression for the particle's velocity v(x) as a function of position at any point x in a fluid whose drag force is expressed as Fdrag = kmv where k is a constant, m is the mass of the particle and v is its velocity. Assume that the particle is constrained to move in the x-axis only with an initial velocity vo- Solution: The net force along the x-axis is: ΣF-F = m then: mv = m Since acceleration is the first time derivative of velocity a = dv/dt, mv = m We can eliminate time dt by expressing, the velocity on the left side of the equation as v = dx/dt. Manipulating the variables and simplifying, we arrive at the following expression = -k "Isolating" the infinitesimal velocity dx and integrating with respect to dx, we arrive at the following: = v -

Problem Like friction, drag force opposes the motion of a particle in a fluid; however, drag force depends on the particle's velocity. Find the expression for the particle's velocity v(x) as a function of position at any point x in a fluid whose drag force is expressed as Fdrag = kmv where k is a constant, m is the mass of the particle and v is its velocity. Assume that the particle is constrained to move in the x-axis only with an initial velocity vo- Solution: The net force along the x-axis is: ΣF-F = m then: mv = m Since acceleration is the first time derivative of velocity a = dv/dt, mv = m We can eliminate time dt by expressing, the velocity on the left side of the equation as v = dx/dt. Manipulating the variables and simplifying, we arrive at the following expression = -k "Isolating" the infinitesimal velocity dx and integrating with respect to dx, we arrive at the following: = v -

Principles of Physics: A Calculus-Based Text

5th Edition

ISBN:9781133104261

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter5: More Applications Of Newton’s Laws

Section: Chapter Questions

Problem 61P

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Like friction, drag force opposes the motion of a particle in a fluid; however, drag force depends on the particle's velocity. Find the expression for the particle's velocity v(x) as a function of position at any point x in a fluid whose drag force is expressed as Fdrag = kmv where k is a constant, m is the mass of the particle and v is its velocity. Assume that the particle is constrained to move in the x-axis only with an initial velocity v0. Solution: The net force along the x-axis is: ΣF = -F = m then: -mv = m Since acceleration is the first time derivative of velocity a = dv/dt, -mv = m We can eliminate time dt by expressing, the velocity on the left side of the equation as v = dx/dt. Manipulating the variables and simplifying, we arrive at the following expression / = -k "Isolating" the infinitesimal velocity dx and integrating with respect to dx, we arrive at the following: = v0 - which shows that velocity decreases in a linear manner.
A dated box of dates, of mass 5.00 kg, is sent sliding up a frictionless ramp at an angle of to the horizontal as a function of time t, the component vx of the box’s velocity along an x axis that extends directly up the ramp.What is the magnitude of the normal force on the box from the ramp?
Recall that Newton’s Equation of Motion for a Body of Mass, m, is F = m(dv/dt). Assume the Initial Velocity of the Projectile is v0, and the Gravitational Acceleration Constant is g, where the Gravitational Force is Fg = -mg. Note that the Maximum Height will Occur at a Time, tm, when the Velocity Goes to Zero, v(tm) = 0. a) Simply Solve the Equation of Motion for the Velocity as a Function of Time, and Determine the Time it Takes the Projectile to Reach the Maximum Height, tm, As a Function of the Given Parameters. b) Include a Force of Air Resistance that is Proportional to the Velocity of the Form Fr = -kmv. Again, Solve the Equation of Motion for the Velocity as a Function of Time, by Direct Integration or by Choosing an Appropriate Ansatz Form for the Solution. Again, Determine an Expression for the Time it Takes the Projectile to Reach the Maximum Height, tm, As a Function of the Given Parameters. c) Finally, Compare the Two Results by First Checking that they Agree for a Small…
The drag force exerted on a car by air depends on a dimensionless drag coefficient, the density of air, the car velocity, and the frontal area of the car. That is, FD = function (CDrag, Afront, r, V). Based on unit considerations alone, obtain a relation for the drag force.
A 4.0 kg object has a velocity of 3i m/s at one instant. Eight seconds later, its velocity is (8i + 10j) m/s. Assuming the object was subject to a constant net force, find (a) the components of the force and (b) its magnitude. show complete solution
A dated box of dates, of mass 7.1 kg, is sent sliding up a frictionless ramp at an angle of θ to the horizontal. The figure here gives, as a function of time t, the component vx of the box's velocity along an x axis that extends directly up the ramp. What is the magnitude of the normal force on the box from the ramp?
A particle of mass m moves along a frictionless, horizontal plane with a speed given byv(x) = a/x, where xis its distance from the origin and ais a positive constant. Find the forceF(s) to which the particle is subject.
In the expression of Newton’s second law for a system of constant mass, F=ma : O The vector representing the net force must pass through the center of mass of the system. O The acceleration term represents the acceleration of a particle located at the center of mass. O The internal forces must be considered when calculating the force term. O The force term can be calculated by considering only the external forces.
Consider two peoplepushing a toboggan with four children on it up a snowcovered slope. Construct a problem in which you calculatethe acceleration of the toboggan and its load. Include a freebody diagram of the appropriate system of interest as thebasis for your analysis. Show vector forces and theircomponents and explain the choice of coordinates. Amongthe things to be considered are the forces exerted by thosepushing, the angle of the slope, and the masses of thetoboggan and children.
Typically, a bullet leaves a standard 45-caliber pistol (5.0-in. barrel) at a speed of 262m/s.If it takes 1 ms to traverse the barrel ,determine the average acceleration experienced by the 16.2g bullet within the gun and then compute the average force exerted on it .
According to legend, Galileo Galilei dropped two balls of different mass from the top of the leaning tower of Pisa in 1589. Whether or not this public experiment ever took place, Galileo was able to demonstrate that, contrary to Aristotle’s teaching, all bodies fall at the same rate regardless of mass, assuming that one is not so tenuous that it is slowed by air resistance. In this experiment, an equation is presented relating the acceleration of gravity at Earth’s surface, g, to the height that an object falls from, h, and the time it takes the object to reach the ground, t. Gravity acceleration at Earth’s surface has been measured many times. In British Imperial Units, Small Metric Units, and Large Metric Units, the standard values of g are: g = 32 feet per second-squared (ft/s2) g = 980 centimeters per second-squared (cm/s2) g = 9.8 meters per second-squared (m/s2). Theory Newton succeeded in explaining gravitational acceleration using his Laws of…
A block, initially at rest, has a mass m and sits on a plane inclined at angle theta. It slides a distance d before hitting a spring and compresses the spring by a maximum distance of xf. If the coefficient of kinetic friction between the plane and block is uk, then what is the force constant of the spring?