Suppose a particle of constant mass m with position x > 0, moves in one space dimension under the influence of the gravitational force of another point particle of constant mass M sitting at x = 0, i.e. the attracting force is F = −GmM/x^2 i (a) Using Newton’s second law, show that d/dt (1/2 mx ̇^2 - −GmM/x) = 0 (i.e., the total energy, sum of kinetic and potential energy, is conserved). (b) Using the change of variables v dv/dx = dv/dt, solve the equation of motion and determine the velocity of the particle v(x) as a function of x assuming it starts with zero velocity at x0. Does the particle’s speed in x = 0 depend on the initial position?
Suppose a particle of constant mass m with position x > 0, moves in one space dimension under the influence of the gravitational force of another point particle of constant mass M sitting at x = 0, i.e. the attracting force is F = −GmM/x^2 i (a) Using Newton’s second law, show that d/dt (1/2 mx ̇^2 - −GmM/x) = 0 (i.e., the total energy, sum of kinetic and potential energy, is conserved). (b) Using the change of variables v dv/dx = dv/dt, solve the equation of motion and determine the velocity of the particle v(x) as a function of x assuming it starts with zero velocity at x0. Does the particle’s speed in x = 0 depend on the initial position?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Suppose a particle of constant mass m with position x > 0, moves in one space dimension under the influence of the gravitational force of another point particle of constant mass M sitting at x = 0, i.e. the attracting force is
F = −GmM/x^2 i
(a) Using Newton’s second law, show that
d/dt (1/2 mx ̇^2 - −GmM/x) = 0
(i.e., the total energy, sum of kinetic and potential energy, is conserved).
(b) Using the change of variables v dv/dx = dv/dt, solve the equation of motion and determine the
velocity of the particle v(x) as a function of x assuming it starts with zero velocity at x0. Does the particle’s speed in x = 0 depend on the initial position?
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