The initial speed that needs to be given to a particle if it is to have a final speed that is equal to its escape speed when it is very far from the Earth.

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Chapter 11, Problem 48P

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The initial speed that needs to be given to a particle if it is to have a final speed that is equal to its escape speed when it is very far from the Earth.

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Question a) A satellite of mass m = 3.6kg orbits the Earth 300km ab ove the Earth'ssurface. How much do es the p otential energy of the satellite change when it islaunched from the surface of the Earth to its orbit? b) By assuming the orbit is circular and the satellite is held in its orbit bygravity, show that the p erio d T of the satellite's orbit can b e expressed as(see image)where r is the distance of the satellite from the Earth's centre and g is theacceleration due to gravity. c) Using the mean radius of the Earth of R = 6400km, calculate the tangentialspeed of the satellite in its orbit. d) If the satellite is launched by ro cket from the Equator, how much do es thekinetic energy of the satellite change when it is placed into this orbit? Do esmost of the energy supplied by the ro cket to the satellite go into the satellite'skinetic energy or the p otential energy?

(a) Calculate how much work is required to launch a spacecraft of mass m from the surface of the earth (mass mE, radius RE) and place it in a circular low earth orbit—that is, an orbit whose altitude above the earth’s surface is much less than RE. (As an example, the International Space Station is in low earth orbit at an altitude of about 400 km, much less than RE = 6370 km.) Ignore the kinetic energy that the spacecraft has on the ground due to the earth’s rotation. (b) Calculate the minimum amount of additional work required to move the spacecraft from low earth orbit to a very great distance from the earth. Ignore the gravitational effects of the sun, the moon, and the other planets. (c) Justify the statement “In terms of energy, low earth orbit is halfway to the edge of the universe.”

a) Derive the escape speed vII (sometimes called the second cosmic speed) from the surface of body of radius R and mass M, using energy conservation. Assume that the planet does not move or spin, so that the total mechanical energy E of a test particle both on the surface and at infinity is equal to zero, and the speed at infinity is zero. Use the formula come up in the last part, Compute this escape speed (in km/s) from Venus, Mars, Jupiter and Neptune. b) Evaluate vIII from the heliocentric orbit of the 4 planets (Venus, Mars, Jupiter and Neptune) mentioned above, assuming circularity of their orbits. Compare the computed speeds with the escape speeds from their surfaces (vII). Notice a big difference in the situation around the inner and the outer planets. What is it due to? A fly-by’s of small bodies near a planet can result in the small particle being accelerated to a final speed close to vII w.r.t. the planet. Draw conclusions as to which planets are able to accelerate small…