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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 GmM X² -i. F= (a) Using Newton's second law, show that d a ( 12m² __ GmM) = dt = 0. (i.e., the total energy, sum of kinetic and potential energy, is conserved). (b) Using the change of variables v , 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 GmM X² -i. F= (a) Using Newton's second law, show that d a ( 12m² __ GmM) = dt = 0. (i.e., the total energy, sum of kinetic and potential energy, is conserved). (b) Using the change of variables v , 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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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 - dt GmM GmM i. = 0. (i.e., the total energy, sum of kinetic and potential energy, is conserved). (b) Using the change of variables v=d, 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
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 (2m²² - dt GmM GmM i. = 0. (i.e., the total energy, sum of kinetic and potential energy, is conserved). (b) Using the change of variables v=d, 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
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