Answer :
Answer:
(a) Ratio of mean density is 0.735
(b) Value of g on mars 0.920 [tex]m,/sec^2[/tex]
(c) Escape velocity on earth is [tex]3.563\times 10^4m/sec[/tex]
Explanation:
We have given radius of mars [tex]R_{mars}=6.9\times 10^3km=6.9\times 10^6m[/tex] and radius of earth [tex]R_{E}=1.3\times 10^4km=1.3\times 10^7m[/tex]
Mass of earth [tex]M_E=5.972\times 10^{24}kg[/tex]
So mass of mars [tex]M_m=5.972\times\times 0.11 \times 10^{24}=0.657\times 10^{24}kg[/tex]
Volume of mars [tex]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[/tex]
So density of mars [tex]d_{mars}=\frac{mass}{volume}=\frac{0.657\times 10^{24}}{1375.357\times 10^{18}}=477.69kg/m^3[/tex]
Volume of earth [tex]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[/tex]
So density of earth [tex]d_{E}=\frac{mass}{volume}=\frac{5.972\times 10^{24}}{9.198\times 10^{21}}=649.271kg/m^3[/tex]
(A) So the ratio of mean density [tex]\frac{d_{mars}}{d_E}=\frac{477.69}{649.27}=0.735[/tex]
(B) Value of g on mars
g is given by [tex]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[/tex]
(c) Escape velocity is given by
[tex]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[/tex]
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.
[tex]\frac{d_{m}}{d_{e}}=\frac{M_{m}}{M_{e}}\times \frac{V_{e}}{V_{m}}[/tex]
[tex]\frac{d_{m}}{d_{e}}=\frac{0.11M_{e}}{M_{e}}\times \frac{1.15\times10^{21}}{1.72\times10^{20}}[/tex]
density of mars : density of earth = 0.72
(b) The value of acceleration due to gravity
[tex]g=\frac{GM}{R^{2}}[/tex]
Let gm be the acceleration due to gravity on Mars
[tex]\frac{g_{m}}{g_{e}}=\frac{M_{m}}{M_{e}}\times \frac{R_{e}^{2}}{R_{m}^{2}}[/tex]
[tex]\frac{g_{m}}{g_{e}}=\frac{0.11M_{e}}{M_{e}}\times \frac{6.5\times6.5}{3.45\times3.45}[/tex]
gm = 3.83 m/s^2
(c) The escape velocity is given by
[tex]v=\sqrt{2gR}[/tex]
[tex]\frac{v_{m}}{v_{e}}=\sqrt{\frac{g_{m}\times R_{m}}{g_{e}\times R_{e}}}[/tex]
[tex]\frac{v_{m}}{v_{e}}=\sqrt{\frac{3.83\times 3.45}{9.8\times 6.5}}[/tex]
escape velocity for mars = 5.1 Km/s