H2. 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: (a) Using Newton's second law, show that d #/(m² GmM) = dt X - GmM -i. = 0. (i.e., the total energy, sum of kinetic and potential energy, is conserved). (b) Using the change of variables v = r, 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?
H2. 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: (a) Using Newton's second law, show that d #/(m² GmM) = dt X - GmM -i. = 0. (i.e., the total energy, sum of kinetic and potential energy, is conserved). (b) Using the change of variables v = r, 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?
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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Transcribed Image Text:H2. 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
(a) Using Newton's second law, show that
d
à (2m²² -
dt
GmM
i.
GmM
x
= 0.
(i.e., the total energy, sum of kinetic and potential energy, is conserved).
(b) Using the change of variables var = r, 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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