Question about acceleration?

Question

Jupiter is the largest planet in our solar system, having a mass and radius that are, respectively, 318 and 11.2 times that of earth. Suppose than an object fallsfrom rest near the surface of each planet and that acceleration due to gravity remains constant during the fall. Each object falls to the same distance. Determine the ratio of the time of fall on Jupiter than that to Earth.

Answer 1

let G,M,R,T be the gravity,mass,radius,time for jupiter
and g,m,r,t be the gravity,mass,radius,time for earth
and let k be the gravitational constant since G is used already
you're given M/m = 318 and R/r = 11.2
then
newtons law
g = km/r²
so
k = gr²/m = constant for both planets
so
gr²/m = GR²/M
gives
G = g(M/m)(r/R)²..............(1)
so you can now find the gravity on jupiter
also, near the surface,
d = ½gt² = constant for both planets
since each object falls the same distance,d.
so
½gt² = ½GT²
gives
T/t
= √(g/G)
= √(g/g(M/m)(r/R)²) ....by (1)
= (R/r)√(m/M)
= 11.2√(1/318)
≈ 0.628

the end
.

Answer 2

Gravity field on a planet is not constant. It is a function of inverse time squared . The Gravitational time time at the surface of the Earth =806 seconds.
The gravitational time at the surface of jupiter is=2.1 time greater than that of the earth.

Answer 3

g(E) = 9.8 m/s² (Acc. due to gravity of Earth)
g(J) = (Acc. due to gravity of Jupiter)
g(J) = (G*M*318) / R² * (11.8)²
. . . . = (G*M / R²)(318 / 11.8²)
. . . .= g(E) * 2.54 = 9.8 * 2.54 = 24.8
The equation we have to use is 'h' = ut + ½gt²
As it is a free fall from rest, u = 0, so 'h' = ½gt²
Or, t² = 2h / g
. . t = √(2h/g).......General equation for time.
t(E) is the time of fall in earth
t(J) is the time of fall in jupiter
g(E) = 9.8 m/s²
g(J) = 24.8 m/s²

t(J) = √(2h/24.8)
t(E) = √(2h/9.8)
{ t(J) / t(E) } = {√(2h/24.8) / √(2h/ 9.8) }
...."...."....."...= √{(2h/24.8) * (9.8/2h)}
The height of fall in both the planets are the same
. . "..."....."...= √(24.8/9.8) = √2.531 = 1.59
Hence, the ratio of time in JUPITER to time in EARTH= 1.59
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