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Answer 1 · 2016-03-28T01:38:46

(a) $0.284 m s^{- 2}$ $0.284ms^-2$ rounded to three decimal places.
(b) $17.89 N$ $17.89N$ rounded to two decimal places
(c) $3.8 s$ $3.8s$ rounded to one decimal place.

Explanation:

As per Law of Universal Gravitation the force of attraction between two bodies is directly proportional to the product of masses of the two bodies. it is also inversely proportional to the square of the distance between the two.
Mathematically
$F_{G} \propto M_{1} . M_{2}$ $F_G prop M_1.M_2$
Also $F_{G} \propto \frac{1}{r^{2}}$ $F_G prop 1/r^2$
Combining the two we obtain the proportionality expression

$F_{G} \propto \frac{M_{1} . M_{2}}{r^{2}}$ $F_G prop (M_1.M_2)/r^2$

$\Rightarrow F_{G} = G \frac{M_{1} . M_{2}}{r^{2}}$ $=>F_G =G (M_1.M_2)/r^2$
Where $G$ $G$ is the proportionality constant.
and has a value $= 6.67408 \times 10^{- 11} m^{3} k g^{- 1} s^{- 2}$ $=6.67408 xx 10^-11 m^3 kg^-1 s^-2$

If one of the bodies is Ceres, and the other body of mass $m$ $m$ the force equation becomes
$F_{Ceres} = G \frac{M_{Ceres} . m}{r_{Ceres}^{2}}$ $F_"Ceres" =G (M_"Ceres".m)/r_"Ceres"^2$
Comparing with the well known equation $F = m \times a or Weight = m \times g$ $F=mxxa or "Weight"=mxx g$
(1) Acceleration due to gravity on the surface of Ceres
$g = G \frac{M_{Ceres}}{r_{Ceres}^{2}}$ $g=G M_"Ceres"/r_"Ceres"^2$ , plug in given values
$g = 6.67408 \times 10^{- 11} \frac{9.4 \times 10^{20}}{{(4.7 \times 10^{5})}^{2}}$ $g=6.67408 xx 10^-11 (9.4xx10^20)/(4.7xx10^5)^2$
$g = 0.284 m s^{- 2}$ $g=0.284ms^-2$ rounded to three decimal places.
(2) Weight of $63 k g$ $63kg$ person on the surface of Ceres $= 63 \times 0.284$ $=63xx0.284$
$= 17.89 N$ $=17.89N$ rounded to two decimal places
(c) We have the following equation for free fall under gravity

$h = u t + \frac{1}{2} g t^{2}$ $h=ut+1/2g t^2$ , plug in known values
$2 = 0 \times t + \frac{1}{2} \times 0.284 \times t^{2}$ $2=0xxt+1/2xx0.284xxt^2$ , solving for $t$ $t$
$t^{2} = 2 \times \frac{2}{0.284}$ $t^2=2xx2/0.284$
$t = 3.8 s$ $t=3.8s$ rounded to one decimal place.

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