74. N In the far future, astronauts travel to the planet and land on Mimas, one

Question

74. N In the far future, astronauts travel to the planet and land on Mimas, one of its 62 moons. Mimas is small compared with the Earth's moon, with mass M 3.75 x 1019 kg and radius R 1.98 x 105 m, giving it a free-fall acceleration of 0.0636 m/s One astronaut, being a baseball fan and hav- ing a strong arm, decides to see how high she can throw a ball in this reduced gravity. She throws the ball straight up the surface of Mimas at a speed of 40 m/s (about 90 mph, the speed of a good major league fastball a. Predict the maximum height of the ball assuming g is constant and using energy conserva- tion. Mimas has no atmosphere, so there is no air resistance b. Now calculate the maximum height using universal gravi- tation. c. How far off is your estimate of part (a)? Express your answer as a percent difference and indicate if the estimate is too high or too low.

Explanation / Answer

Given ,
U = 40m/s
g = 0.0636 m/s^2
G = 6.673 * 10^-11 N m^2/Kg^2
Mass of Planet M = 3.75 * 10^19
Radius of Planet r = 1.98 *10^5

a)
Assuming g to be constant and using energy conservation -
We get

u^2 = 2 * g * h
40^2 = 2 * 0.0636 * h
h = 12578.61 m

B)
According to Universal Gravtiation method -
- GMm /r + 0.5*m*v^2 = - GMm/(r+h)
(- 6.673 * 10^-11 * 3.75 * 10^19) / (1.98 *10^5) + 0.5 * 40^2 = (- 6.673 * 10^-11 * 3.75 * 10^19) / (1.98 *10^5 + h)

Solving for h

h = 13380.3 m

c)

Difference in estimate = (13380.3 - 12578.61)/13380.3 * 100%
Difference in estimate = 5.99%

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