Time period of any moon of Jupiter is given by
![T = 2\pi \sqrt{\frac{r^3}{GM}}](https://tex.z-dn.net/?f=T%20%3D%202%5Cpi%20%5Csqrt%7B%5Cfrac%7Br%5E3%7D%7BGM%7D%7D)
from above formula we can say that mass of Jupiter is given by
![M = \frac{4 \pi^2 r^3}{GT^2}](https://tex.z-dn.net/?f=M%20%3D%20%5Cfrac%7B4%20%5Cpi%5E2%20r%5E3%7D%7BGT%5E2%7D)
now for part a)
![r = 4.22 * 10^8 m](https://tex.z-dn.net/?f=r%20%3D%204.22%20%2A%2010%5E8%20m)
T = 1.77 day = 152928 seconds
now by above formula
![M = \frac{4 \pi^2 r^3}{GT^2}](https://tex.z-dn.net/?f=M%20%3D%20%5Cfrac%7B4%20%5Cpi%5E2%20r%5E3%7D%7BGT%5E2%7D)
![M = \frac{4 \pi^2 (4.22 * 10^8)^3}{(6.67 * 10^{-11})(152928)^2}](https://tex.z-dn.net/?f=M%20%3D%20%5Cfrac%7B4%20%5Cpi%5E2%20%284.22%20%2A%2010%5E8%29%5E3%7D%7B%286.67%20%2A%2010%5E%7B-11%7D%29%28152928%29%5E2%7D)
![M = 1.9* 10^{27} kg](https://tex.z-dn.net/?f=M%20%3D%201.9%2A%2010%5E%7B27%7D%20kg)
Part B)
![r = 6.71 * 10^8 m](https://tex.z-dn.net/?f=r%20%3D%206.71%20%2A%2010%5E8%20m)
T = 3.55 day = 306720 seconds
now by above formula
![M = \frac{4 \pi^2 r^3}{GT^2}](https://tex.z-dn.net/?f=M%20%3D%20%5Cfrac%7B4%20%5Cpi%5E2%20r%5E3%7D%7BGT%5E2%7D)
![M = \frac{4 \pi^2 (6.71 * 10^8)^3}{(6.67 * 10^{-11})(306720)^2}](https://tex.z-dn.net/?f=M%20%3D%20%5Cfrac%7B4%20%5Cpi%5E2%20%286.71%20%2A%2010%5E8%29%5E3%7D%7B%286.67%20%2A%2010%5E%7B-11%7D%29%28306720%29%5E2%7D)
![M = 1.9* 10^{27} kg](https://tex.z-dn.net/?f=M%20%3D%201.9%2A%2010%5E%7B27%7D%20kg)
Part c)
![r = 10.7 * 10^8 m](https://tex.z-dn.net/?f=r%20%3D%2010.7%20%2A%2010%5E8%20m)
T = 7.16 day = 618624 seconds
now by above formula
![M = \frac{4 \pi^2 r^3}{GT^2}](https://tex.z-dn.net/?f=M%20%3D%20%5Cfrac%7B4%20%5Cpi%5E2%20r%5E3%7D%7BGT%5E2%7D)
![M = \frac{4 \pi^2 (10.7 * 10^8)^3}{(6.67 * 10^{-11})(618624)^2}](https://tex.z-dn.net/?f=M%20%3D%20%5Cfrac%7B4%20%5Cpi%5E2%20%2810.7%20%2A%2010%5E8%29%5E3%7D%7B%286.67%20%2A%2010%5E%7B-11%7D%29%28618624%29%5E2%7D)
![M = 1.89* 10^{27} kg](https://tex.z-dn.net/?f=M%20%3D%201.89%2A%2010%5E%7B27%7D%20kg)
PART D)
![r = 18.8 * 10^8 m](https://tex.z-dn.net/?f=r%20%3D%2018.8%20%2A%2010%5E8%20m)
T = 16.7 day = 1442880 seconds
now by above formula
![M = \frac{4 \pi^2 r^3}{GT^2}](https://tex.z-dn.net/?f=M%20%3D%20%5Cfrac%7B4%20%5Cpi%5E2%20r%5E3%7D%7BGT%5E2%7D)
![M = \frac{4 \pi^2 (18.8 * 10^8)^3}{(6.67 * 10^{-11})(1442880)^2}](https://tex.z-dn.net/?f=M%20%3D%20%5Cfrac%7B4%20%5Cpi%5E2%20%2818.8%20%2A%2010%5E8%29%5E3%7D%7B%286.67%20%2A%2010%5E%7B-11%7D%29%281442880%29%5E2%7D)
![M = 1.889* 10^{27} kg](https://tex.z-dn.net/?f=M%20%3D%201.889%2A%2010%5E%7B27%7D%20kg)
Answer:
The capacite is C=5.32 uF using the equations of voltage and energy in capacitance
Explanation:
The energy holds is 5 J and the resistor dissipates 2J so the energy total is 3J
Using:
![V_{t}= V_{o}e^{\frac{-t}{R*C} }](https://tex.z-dn.net/?f=V_%7Bt%7D%3D%20V_%7Bo%7De%5E%7B%5Cfrac%7B-t%7D%7BR%2AC%7D%20%7D)
Voltage in this case is the energy dissipated so
![E_{t}= E_{o}e^{\frac{-t}{R*C} }](https://tex.z-dn.net/?f=E_%7Bt%7D%3D%20E_%7Bo%7De%5E%7B%5Cfrac%7B-t%7D%7BR%2AC%7D%20%7D)
![\frac{\sqrt{E_t} }{\sqrt{E_o} } = e^{\frac{-t}{R*C} }](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csqrt%7BE_t%7D%20%7D%7B%5Csqrt%7BE_o%7D%20%7D%20%3D%20e%5E%7B%5Cfrac%7B-t%7D%7BR%2AC%7D%20%7D)
![\frac{\sqrt{3 J} }{\sqrt{5J} } = e^{\frac{-13.6ms}{10kw*C} }](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csqrt%7B3%20J%7D%20%7D%7B%5Csqrt%7B5J%7D%20%7D%20%3D%20e%5E%7B%5Cfrac%7B-13.6ms%7D%7B10kw%2AC%7D%20%7D)
Using the equation to find capacitance
![ln 0.775= e^{\frac{-13.6 x10^{3} }{10x10^{3}*C }} \\ln(0.775)= ln * e^{\frac{-13.6 x10^{3s} }{10x10^{3}*C }} \\\\ln(0.775)= {\frac{-13.6 x10^{3} }{10x10^{3}*C }} \\C= \frac{-13.6 x10^{-3} }{10x10x^{3}*ln(0.775) }](https://tex.z-dn.net/?f=ln%200.775%3D%20e%5E%7B%5Cfrac%7B-13.6%20x10%5E%7B3%7D%20%7D%7B10x10%5E%7B3%7D%2AC%20%7D%7D%20%5C%5Cln%280.775%29%3D%20ln%20%2A%20e%5E%7B%5Cfrac%7B-13.6%20x10%5E%7B3s%7D%20%7D%7B10x10%5E%7B3%7D%2AC%20%7D%7D%20%5C%5C%5C%5Cln%280.775%29%3D%20%7B%5Cfrac%7B-13.6%20x10%5E%7B3%7D%20%7D%7B10x10%5E%7B3%7D%2AC%20%7D%7D%20%5C%5CC%3D%20%5Cfrac%7B-13.6%20x10%5E%7B-3%7D%20%7D%7B10x10x%5E%7B3%7D%2Aln%280.775%29%20%7D)
F
C= 5.32 uF because u is the symbol for micro that is equal to ![10^{-6}](https://tex.z-dn.net/?f=10%5E%7B-6%7D)
"Voltage" is the "pressure" that makes electrons want to leave where they are
and head in some direction, if there's conducting material in that direction.
"Current" is the rate at which they all migrate in that direction.
The time the package travels horizontally is equal to the time it takes to hit the ground. This can be calculated using:
s = ut + 1/2 at²; u is 0
480 = 4.9t²
t = 9.90 seconds
Horizontal distance = horizontal speed x time
The speed will be converted to m/s from km/h
= 180 km/hr x 1000m/km x 1hr/3600 seconds x 9.90 seconds
= 495 m