Answer:
![\Delta t= 2962.395\,s\,(49.373\,min)](https://tex.z-dn.net/?f=%5CDelta%20t%3D%202962.395%5C%2Cs%5C%2C%2849.373%5C%2Cmin%29)
Explanation:
Let assume that cooling water works under a pressure of 1 atmosphere. The time required to boil half of the water is determined by the First Law of Thermodynamics:
![\dot Q \cdot \Delta t = m \cdot [c_{p,w}\cdot (T_{2}-T_{1})+h_{fg}]](https://tex.z-dn.net/?f=%5Cdot%20Q%20%5Ccdot%20%5CDelta%20t%20%3D%20m%20%5Ccdot%20%5Bc_%7Bp%2Cw%7D%5Ccdot%20%28T_%7B2%7D-T_%7B1%7D%29%2Bh_%7Bfg%7D%5D)
![\Delta t = \frac{m\cdot [c_{p,w}\cdot (T_{2}-T_{1})+h_{fg}]}{\dot Q}](https://tex.z-dn.net/?f=%5CDelta%20t%20%3D%20%5Cfrac%7Bm%5Ccdot%20%5Bc_%7Bp%2Cw%7D%5Ccdot%20%28T_%7B2%7D-T_%7B1%7D%29%2Bh_%7Bfg%7D%5D%7D%7B%5Cdot%20Q%7D)
![\Delta t = \frac{(2.25\times 10^{5}\,kg)\left[\left(4.186\,\frac{kJ}{kg\cdot ^{\textdegree}C} \right)\cdot (100\,^{\textdegree}C - 10\,^{\textdegree}C)+2256.5\,\frac{kJ}{kg} \right]}{200000\,kW}](https://tex.z-dn.net/?f=%5CDelta%20t%20%3D%20%5Cfrac%7B%282.25%5Ctimes%2010%5E%7B5%7D%5C%2Ckg%29%5Cleft%5B%5Cleft%284.186%5C%2C%5Cfrac%7BkJ%7D%7Bkg%5Ccdot%20%5E%7B%5Ctextdegree%7DC%7D%20%5Cright%29%5Ccdot%20%28100%5C%2C%5E%7B%5Ctextdegree%7DC%20-%2010%5C%2C%5E%7B%5Ctextdegree%7DC%29%2B2256.5%5C%2C%5Cfrac%7BkJ%7D%7Bkg%7D%20%5Cright%5D%7D%7B200000%5C%2CkW%7D)
![\Delta t= 2962.395\,s\,(49.373\,min)](https://tex.z-dn.net/?f=%5CDelta%20t%3D%202962.395%5C%2Cs%5C%2C%2849.373%5C%2Cmin%29)
It's implying that a constant force was applied throughout, which also means that there was a constant, negative acceleration. Because of that, the deceleration is equal to the change in velocity (-24.3 m/s) divided by the time (17.2 s):
![\frac{-24.3 m/s}{17.2 s}](https://tex.z-dn.net/?f=%20%5Cfrac%7B-24.3%20m%2Fs%7D%7B17.2%20s%7D%20)
=-1.41279 m/s^2. The force is technically negative because it is causing a negative acceleration. Because F=ma: -2170kg m/sec^2=-(-1.41279 m/sec^2)m. m=
![\frac{-2170 kg m/s^2}{-1.41279 m/sec^2}](https://tex.z-dn.net/?f=%20%5Cfrac%7B-2170%20kg%20m%2Fs%5E2%7D%7B-1.41279%20m%2Fsec%5E2%7D%20)
=<u>1535.9678 kg</u>
Answer:
The velocity of the ions relative to the earth observer is ![v = 0.63 c](https://tex.z-dn.net/?f=v%20%3D%200.63%20c)
Explanation:
From the question we are told that
The speed of the alien ship is ![v_a = 0.3c](https://tex.z-dn.net/?f=v_a%20%20%3D%200.3c)
The speed of the iron gun is ![v_g = 0.4c](https://tex.z-dn.net/?f=v_g%20%20%3D%20%200.4c)
The motion of the ions relative to the earths observer can be mathematically represented as
![v = \frac{v_a + v_g}{[\frac{1+ v_av_g}{c^2} ]}](https://tex.z-dn.net/?f=v%20%20%3D%20%20%5Cfrac%7Bv_a%20%2B%20v_g%7D%7B%5B%5Cfrac%7B1%2B%20v_av_g%7D%7Bc%5E2%7D%20%5D%7D)
substituting values
![v = \frac{ 0.3c + 0.4c}{[\frac{1+ 0.3c * 0.4c}{c^2} ]}](https://tex.z-dn.net/?f=v%20%20%3D%20%20%5Cfrac%7B%200.3c%20%2B%200.4c%7D%7B%5B%5Cfrac%7B1%2B%200.3c%20%2A%200.4c%7D%7Bc%5E2%7D%20%5D%7D)
=> ![v = 0.63 c](https://tex.z-dn.net/?f=v%20%3D%200.63%20c)
1 mile = 5,280 feet
5,280 feet = 63,360 inches
63,360 inches = 160,934.4 centimeters