6-12. Geysers have been observed on several bodies within the Solar System. Wate

Question

6-12. Geysers have been observed on several bodies within the Solar System. Water shot up by Old Faithful in Yellowstone Park reaches a height of up to 50 m. The Prometheus plume on Io is 60 km high, much of the water from Enceladus's south polar plume escapes the moon entirely and plumes on Triton reach a height of 8 km. (a) Assuming that the heights of each of the above geysers are controlled only by ballistics, compute the exit velocity for each geyser as the erupted material leaves the vent. Express your answer both as a velocity and as a ratio to each world's escape velocity (b) *Discuss what your answers may be telling you about the fundamental mecha- nism behind each type of plume and con- sider where this simple ballistic assump- tion may be wrong.

Explanation / Answer

Using equation of motion for the question:

V2 = U2 + 2 x a x s

Where: V= final velocity (It would be zero at maximum height)

U = initial velocity

a = acceleration due to gravity (in this case it would be retardation due to gravity and its value would be -9.81 m/s2)

s = height attained by the geyser

For Yellowstone:

0 = U2 + 2x(-9.81)x50

981 = U2

U = 31.32 meter/second = exit velocity

Earth escape velocity = 11200 m/s

Ratio = 31.32/11200 = 2.76 x 10-3

For Prometheus:

0 = U2 + 2x(-9.81)x60x1000 (there are 1000 meter in 1 kilometer)

U2 = 1,177,200

U = 1,177,200 meter/second = exit velocity

Prometheus excape velocity = 19 m/s

Ratio = 1,177,200/19 =61,957.89

For triton:

0 = U2 + 2x(-9.81)x8x1000 (there are 1000 meter in 1 kilometer)

U2 = 156,960

U = 396.18 meter/second = exit velocity

Triton excape velocity = 1455 m/s

Ratio = 396.18/1455 = 0.2722

(b) The above resuts show that the heavenly bodies where the excape velocity is low there the plume would be going to higher heights because there would be low acceleration acceleration due to gravity. The assumption would be wrong because we are considering the acceleration due to gravity that is present on the earth.

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