Answer:
Diameter, d = 4237.14 meters
Explanation:
It is given that,
On earth you can jump straight up a distance of 36 cm, h = 36 cm = 0.36 m
Asteroids are made of material with mass density, [tex]d=2800\ kg/m^3[/tex]
Escape velocity is given by :
[tex]v=\sqrt{\dfrac{2GM}{R}}[/tex]
[tex]R=\dfrac{2GM}{v^2}[/tex]
Where
G is the universal gravitational constant
M is the mass
Density, [tex]d=\dfrac{M}{V}[/tex]
Density, [tex]d=\dfrac{M}{4/3 \pi R^3}[/tex]
[tex]M=2800\times \dfrac{4}{3}\pi \times R^3=11728.61\ R^3[/tex]..............(2)
Now using conservation of energy as :
[tex]\dfrac{1}{2}mv^2=mgh[/tex]
[tex]v=\sqrt{2gh} =\sqrt{2\times 9.8\times 0.36}=2.65\ m/s[/tex]
[tex]R=\dfrac{2\times 6.67\times 10^{-11}\times 11728.61\ R^3}{(2.65)^2}[/tex]
[tex]R^2=\dfrac{(2.65)^2}{2\times 6.67\times 10^{-11}\times 11728.61}[/tex]
Radius, R = 2118.57 m
Diameter, d = 2R
[tex]d=2\times 2118.57=4237.14\ m[/tex]
So, the maximum diameter of a spherical asteroid from which you could escape by jumping is 4237.14 meters. Hence, this is the required solution.
