H2. Suppose a particle of constant mass m with position x > 0, moves in one space dimensionunder the influence of the gravitational force of another point particle of constant mass Msitting at x = 0, i.e. the attracting force isF=(a) Using Newton's second law, show thatGmMx²i.1dr (2m²2² _ GimM)dt= 0.(i.e., the total energy, sum of kinetic and potential energy, is conserved).dx dt(b) Using the change of variables vdy = r, solve the equation of motion and determine thevelocity of the particle v(x) as a function of x assuming it starts with zero velocity at xo. Doesthe particle's speed in x = 0 depend on the initial position?

Given: F=-GmMx2i Solution: Let the velocity has the velocity ,v=vi From the Newton's second law of…

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 (2m²² -mx dt GmM x² = GmM = 0. (i.e., the total energy, sum of kinetic and potential energy, is conserved). (b) Using the change of variables vd dr, 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? X

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 (2m²² -mx dt GmM x² = GmM = 0. (i.e., the total energy, sum of kinetic and potential energy, is conserved). (b) Using the change of variables vd dr, 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? X

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Concept explainers

Conservation of Energy

In physics, the rule of the conservation of energy states that energy doesn’t disappear from the atmosphere; instead, it changes its form from one type to another. Energy is one of the most important physical quantities in nature, and it cannot be created or destroyed — only transformed.

Question

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
GmM
x²
i.
1
dr (2m²2² _ GimM)
dt
= 0.
(i.e., the total energy, sum of kinetic and potential energy, is conserved).
dx dt
(b) Using the change of variables vdy = 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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