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2. 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 x2 i. (a) Using Newton's second law, show that d 1 dt (/m²² GmM) = = 0. (i.e., the total energy, sum of kinetic and potential energy, is conserved). dv dt (b) Using the change of variables vd = du, 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 xo. Does the particle's speed in x = 0 depend on the initial position?

2. 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 x2 i. (a) Using Newton's second law, show that d 1 dt (/m²² GmM) = = 0. (i.e., the total energy, sum of kinetic and potential energy, is conserved). dv dt (b) Using the change of variables vd = du, 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 xo. Does the particle's speed in x = 0 depend on the initial position?

University Physics Volume 3
University Physics Volume 3
17th Edition
ISBN:9781938168185
Author:William Moebs, Jeff Sanny
Publisher:William Moebs, Jeff Sanny
Chapter11: Particle Physics And Cosmology
Section: Chapter Questions
Problem 69P: Show that the velocity of a star orbiting its galaxy in a circular oibit is inversely proportional...
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2. 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 x2 i. (a) Using Newton's second law, show that d 1 dt (/m²² GmM) = = 0. (i.e., the total energy, sum of kinetic and potential energy, is conserved). dv dt (b) Using the change of variables vd = du, 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 xo. Does the particle's speed in x = 0 depend on the initial position?
Transcribed Image Text:2. 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 x2 i. (a) Using Newton's second law, show that d 1 dt (/m²² GmM) = = 0. (i.e., the total energy, sum of kinetic and potential energy, is conserved). dv dt (b) Using the change of variables vd = du, 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 xo. Does the particle's speed in x = 0 depend on the initial position?
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