Universe
Universe
11th Edition
ISBN: 9781319039448
Author: Robert Geller, Roger Freedman, William J. Kaufmann
Publisher: W. H. Freeman
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Chapter 4, Problem 50Q
To determine

The escape speed from the surfaces of Earth and Mars. State using these values whether it is easier for a rock to escape from Mars or Earth. Use the values provided in Appendix 2.

Expert Solution & Answer
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Answer to Problem 50Q

Solution:

The escape speed from the surface of Earth is 11.2 km/s and that from Mars is 5 km/s. It would be easier for the rock to escape from Mars.

Explanation of Solution

Given data:

Refer to Appendix 2 for the following values:

The equatorial diameter of Earth is 12756km and that of Mars is 6794km.

The mass of Earth is 5.974×1024 kg and that of Mars is 6.418×1023 kg.

Formula used:

State the expression for escape velocity.

V=2GMR

Here, V is the escape velocity, G is the constant of gravitation, M is the mass of the planet, and R is the radius of the planet.

Explanation:

It is known that the radius of a sphere is half of its diameter. Consider Earth to be spherical in shape and calculate its radius as,

Re=De2

Here, subscript e refers to the corresponding quantities for Earth.

Substitute 12756km for De.

Re=12756km2=6378 km(1000 m1 km)=6378000 m

Consider Mars to be spherical in shape and calculate its radius as,

Rm=Dm2

Here, subscript m refers to the corresponding quantities for Mars.

Substitute 6794km for Dm.

Rm=6794km2=3397 km(1000 m1 km)=3397000 m

State the expression for the escape velocity for Earth.

Ve=2GMeRe

Here, subscript e refers to the corresponding quantities for Earth.

Substitute 6.67×1011m3kg1s2 for G, 6378000 m for Re, and 5.974×1024 kg for Me.

Ve=(2)(6.67×1011m3kg1s2)(5.974×1024 kg)(6378000 m)=11178m/s(1 km/s1000 km/s)11.2 km/s

State the expression for the escape velocity for Mars.

Vm=2GMmRm

Here, subscript m refers to the corresponding quantities for Mars.

Substitute 6.67×1011m3kg1s2 for G, 3397000 m for Rm, and 6.418×1023 kg for Mm.

Vm=(2)(6.67×1011m3kg1s2)(6.418×1023 kg)(3397000 m)=5020.3m/s(1 km/s1000 km/s)5 km/s

Conclusion:

Since the escape speed from the surface of Mars is lower than that from Earth, it would be easier for the rock to escape from Mars.

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