The maximum vertical distance through which a full dressed astronaut can jump on the earth is 0.5m. Estimate the maximum vertical distance through which he can jump on the moon, which has a mean density 2/3rd that of earth and radius one quarter that of the earth.
Answers
The required equation to solve this problem :- v2 = u2-2gh . v is final speed in this case it is zero, when astronaut jumps and reaches maximum height his speed is zero. u is initial jumping speed. g is acceleration due to gravity and h is the maximum jumping height.
If jumping speed on earth and moon are assumed as same, then height is inversely proportional to acceleration due to gravity.
begin mathsize 12px style h proportional to 1 over g end style
now acceleration due to gravity begin mathsize 12px style g space equals space fraction numerator G M over denominator R squared end fraction space equals space G over R squared 4 over 3 pi R cubed rho space equals space G 4 over 3 pi R rho end style
G - gravitational constant M mass of earth, ρ is mean density, R is radius
If we consider 2/3 of earth density as moon's density and 1/2 of earth radius as moon's radius
then acceleration due to gravity on moon, begin mathsize 12px style g subscript M space equals 1 over 6 g subscript E end style
gE is acceleration due to gravity on earth
begin mathsize 12px style fraction numerator j u m p i n g space h e i g h t space o n space M o o n over denominator j u m p i n g space h e i g h t space o n space E a r t h end fraction space equals space g subscript E over g subscript M space equals space fraction numerator g subscript E over denominator open parentheses begin display style bevelled 1 over 6 end style close parentheses begin display style begin display style g end style subscript E end style end fraction space equals space 6 end style
Hence the astronaut will jump a height of 6 × 0.5 = 3 m on moon