Particles 1 and 2 of equal mass are thrown vertically upwards at the same initial velocity vo in a constant gravitational field. Particle 1 is under a negligible air resistance fr = 0 while Particle 2 experiences a resistance of the form fr = mov, where m is the particle's mass, a is a positive constant, and v is the particle's velocity at any point in time. Show that the ratio (t2/t1) of the times required for the particles to reach maximum height is given by t2 = 1- avo + O 2g avo ti 2g where O denotes higher order terms. In the limit a → 0, your result for t2 must approach t1.

Particles 1 and 2 of equal mass are thrown vertically upwards at the same initial velocity vo in a constant gravitational field. Particle 1 is under a negligible air resistance fr = 0 while Particle 2 experiences a resistance of the form fr = mov, where m is the particle's mass, a is a positive constant, and v is the particle's velocity at any point in time. Show that the ratio (t2/t1) of the times required for the particles to reach maximum height is given by t2 = 1- avo + O 2g avo ti 2g where O denotes higher order terms. In the limit a → 0, your result for t2 must approach t1.

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter2: Newtonian Mechanics-single Particle

Section: Chapter Questions

Problem 2.9P: Consider a projectile fired vertically in a constant gravitauonal field. For the same initial...

See similar textbooks

Similar questions

A particle of mass 2.4 kg is subject to a force that is always pointed towards the North but whose magnitude changes quadratically with time. Let the y-axis point towards the North. The magnitude of the force is given as F = 6t2, and has units of newtons Determine the change in velocity Δv, in meters per second, of the particle between t=0 and t=1.3s. Determine the change in y coordinate, in meters, of the particle Δ y betweent=o and t=1.3 if the initial velocity= 15.9 m/s and directed North, in the same direction as the force.
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…
Suppose that a projectile of mass m moves in a verticalplane in the atmosphere near the surface of the earth underthe influence of two forces: a downward gravitationalforce of magnitude mg, and a resistive force FR that isdirected opposite to the velocity vector v and has magnitudek v2 (where v = |v| is the speed of the projectile;see Fig. 4.1.15). Show that the equations of motion of theprojectile are mx'' = -kvx' , my'' = -kvy' -mg,
A 3.5 kg particle moves along an x axis according to x(t) = −12 + 5t2 − 4t3, with xin meters and t in seconds. In unit-vector notation, what is the net force acting on theparticle at t = 6 s?
A particle of mass 1.05 kg is subject to a force that is always pointed towards the East but whose magnitude changes linearly with time t. The magnitude of the force is given as F = 4t, and has units of newtons. Let the x-axis point towards the East. Determine the change in x-coordinate in meters of the particle Δx between t = 0 and t = 2.4 if the initial velocity is 17.5 m/s, and pointed in the same direction as the force.
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?
Let us make the (unrealistic) assumption that a boat of mass m gliding with initial velocity v0 in water is slowed by a viscous retarding force of magnitude bv2, where b is a constant, (a) Find and sketch v(t). How long does it take the boat to reach a speed of v0/l000? (b) Find x(t). How far does the boat travel in this time? Let m = 200 kg, v0 = 2 m/s, and b = 0.2 Nm-2s2.
Consider a projectile fired vertically in a constant gravitauonal field. For the same initial velocities, compare the times required the projectile to reach its maximum height (a) for zero resisting force, (b) for a resisting force proportional to the instantaneous velocity of the projectile.
If the vector components of the position of a particle moving in the xy plane as a function of time are x(t)=(2.5ms2)t2i and y(t)=(5.0ms3)t3j, when is the angle between the net force on the particle and the x axis equal to 45?
A particle of mass 0.75 kg is subject to a force that is always pointed towards the East but whose magnitude changes linearly with time t. The magnitude of the force is given as F = 5t, and has units of newtons. Let the x-axis point towards the East. Determine the change in the velocity Δv, in meters per second, of the particle between t = 0 and t = 1.2 sec. Determine the change in x-coordinate in meters of the particle Δx between t = 0 and t = 1.2 if the initial velocity is 13.4 m/s, and pointed in the same direction as the force.
A Block of mass m is moving to the left with a velocity of magnitude v1. At t=0 a force is exerted on the block, pointing to the right with magnitude c1t where c1 is a known constant. How far will the block travel to the left before it reverses its direction of motion
A 2.0 kg projectile with initial velocity v→ = 6.5i m/s experiences the variable force F→ = -2.0ti +4.0t2j N, where t is in s. What is the projectile's speed at t = 2.0 s? Also, at what instant of time is the projectile moving parallel to the y-axis?