Question

The mean diameters of Mars and Earth are 6.9 ✕ 103 km and 1.3 ✕ 104 km, respectively. The mass of Mars is 0.11 times Earth's mass.
(a) What is the ratio of the mean density of Mars to that of Earth?
(b) What is the value of g on Mars? m/s2
(c) What is the escape speed on Mars? m/s2

Answer 1

Answer:

(a) 0.72

(b) 3.83 m/s^2

(c)  5.1 Km/s

Explanation:

diameter of Mars = 6.9 x 10^3 km

Radius of Mars, Rm = 3.45 x 10^3 km = 3.45 x 10^6 m

diameter of earth = 1.3 x 10^4 km

radius of earth, Re = 6.5 x 10^3 km = 6.5 x 10^6 m

Let Me be the mass of earth.

Mass of Mars, Mm = 0.11 Me

(a) Volume of Mars, Vm = 4/3 x 3.14 x (3.45 x 10^6)³ = 1.72 x 10^20 m³

Volume of earth, Ve =  4/3 x 3.14 x (6.5 x 10^6)³ = 1.15 x 10^21 m³

density is the ratio of mass to the volume of the object.

$\frac{d_{m}}{d_{e}}=\frac{M_{m}}{M_{e}}\times \frac{V_{e}}{V_{m}}$

$\frac{d_{m}}{d_{e}}=\frac{0.11M_{e}}{M_{e}}\times \frac{1.15\times10^{21}}{1.72\times10^{20}}$

density of mars : density of earth = 0.72

(b) The value of acceleration due to gravity

$g=\frac{GM}{R^{2}}$

Let gm be the acceleration due to gravity on Mars

$\frac{g_{m}}{g_{e}}=\frac{M_{m}}{M_{e}}\times \frac{R_{e}^{2}}{R_{m}^{2}}$

$\frac{g_{m}}{g_{e}}=\frac{0.11M_{e}}{M_{e}}\times \frac{6.5\times6.5}{3.45\times3.45}$

gm = 3.83 m/s^2

(c) The escape velocity is given by

$v=\sqrt{2gR}$

$\frac{v_{m}}{v_{e}}=\sqrt{\frac{g_{m}\times R_{m}}{g_{e}\times R_{e}}}$

$\frac{v_{m}}{v_{e}}=\sqrt{\frac{3.83\times 3.45}{9.8\times 6.5}}$

escape velocity for mars = 5.1 Km/s

Answer 2

Answer:

(a) Ratio of mean density is 0.735

(b) Value of g on mars 0.920 $m,/sec^2$

(c) Escape velocity on earth is $3.563\times 10^4m/sec$

Explanation:

We have given radius of mars $R_{mars}=6.9\times 10^3km=6.9\times 10^6m$ and radius of earth $R_{E}=1.3\times 10^4km=1.3\times 10^7m$

Mass of earth $M_E=5.972\times 10^{24}kg$

So mass of mars $M_m=5.972\times\times 0.11 \times 10^{24}=0.657\times 10^{24}kg$

Volume of mars $V=\frac{4}{3}\pi R^3=\frac{4}{3}\times 3.14\times (6.9\times 10^6)^3=1375.357\times 10^{18}m^3$

So density of mars $d_{mars}=\frac{mass}{volume}=\frac{0.657\times 10^{24}}{1375.357\times 10^{18}}=477.69kg/m^3$

Volume of earth  $V=\frac{4}{3}\pi R^3=\frac{4}{3}\times 3.14\times (1.3\times 10^7)^3=9.198\times 10^{21}m^3$

So density of earth $d_{E}=\frac{mass}{volume}=\frac{5.972\times 10^{24}}{9.198\times 10^{21}}=649.271kg/m^3$

(A) So the ratio of mean density $\frac{d_{mars}}{d_E}=\frac{477.69}{649.27}=0.735$

(B) Value of g on mars

g is given by $g=\frac{GM}{R^2}=\frac{6.67\times 10^{-11}\times0.657\times 10^{24}}{(6.9\times 10^6)^2}=0.920m/sec^2$

(c) Escape velocity is given by

$v=\sqrt{\frac{2GM}{R}}=\sqrt{\frac{2\times 6.67\times 10^{-11}\times 0.657\times 10^{24}}{6.9\times 10^6}}=3.563\times 10^4m/sec$