A satellite made of aluminum with mass 3500 kg travels on circulated path in 100 km height over the earth. Because of airresistance of the airveil the satellite loses energy and falls in a spiral down to earth. a) What is the mechanical energy of the satellite in the beginning and when he hits the earth. What is the change of the mechanical energy? b) If the whole mechanical energy transformes to heat, can the heat of the satellite get the melting point of aluminum? (Here you have to find heat capacity and the melting point of aluminum and use equation: Q=mcΔT).
a) mechanical energy= PE+KE
PE= m(mearth*g*(Re/(Re+altitude)) *^2* altitude
KE= 1/2 m v^2 where v is found by
mv^2/(Re+h)= PE/h
solve for v.
b) mechanical energy at end? zero
To answer these questions, we'll need to consider the gravitational potential energy and the kinetic energy of the satellite.
a) In the beginning, the satellite has gravitational potential energy and kinetic energy.
The gravitational potential energy (PE) is given by:
PE = mgh
where:
m = mass of the satellite = 3500 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height of the satellite above the Earth's surface
Since the satellite is at a height of 100 km, which is equivalent to 100,000 meters, we can calculate the gravitational potential energy:
PE = 3500 kg * 9.8 m/s^2 * 100,000 m = 3.43 x 10^9 J
The kinetic energy (KE) of the satellite is given by:
KE = (1/2)mv^2
where:
v = velocity of the satellite
Since the satellite is traveling in a circular path, we can use the equation relating velocity, radius (height), and acceleration:
v = √(gR)
where:
R = radius of the circular path = height of the satellite above the Earth's surface + radius of the Earth
Considering the Earth's radius is approximately 6,371 km (which is 6,371,000 meters), we can calculate the velocity:
v = √(9.8 m/s^2 * (100,000 m + 6,371,000 m)) = 7,904 m/s
Now we can calculate the kinetic energy:
KE = (1/2) * 3500 kg * (7904 m/s) ^ 2 = 9.75 x 10^10 J
The total mechanical energy (ME) of the satellite in the beginning is the sum of the gravitational potential energy and the kinetic energy:
ME = PE + KE = 3.43 x 10^9 J + 9.75 x 10^10 J = 1.03 x 10^11 J
When the satellite hits the Earth, the height (h) becomes zero, and therefore, the gravitational potential energy (PE) also becomes zero. The mechanical energy (ME) at this point is equal to the kinetic energy (KE):
ME = KE = (1/2)mv^2 = (1/2) * 3500 kg * (7904 m/s) ^ 2 = 9.75 x 10^10 J
The change in mechanical energy is given by the difference between the initial and final mechanical energies:
Change in ME = ME_final - ME_initial = 9.75 x 10^10 J - 1.03 x 10^11 J = -5.54 x 10^9 J (negative sign indicating a decrease in energy)
b) To determine if the heat generated by the satellite falling to Earth can reach the melting point of aluminum, we need to calculate the amount of heat produced and compare it to the heat required to melt aluminum.
The heat (Q) can be calculated using the equation:
Q = mcΔT
where:
m = mass of the satellite = 3500 kg
c = specific heat capacity of aluminum = 0.897 J/g°C
ΔT = change in temperature
To find the change in temperature (ΔT), we need to know the initial and final temperatures. Let's assume the satellite starts at room temperature (25°C) and the melting point of aluminum is approximately 660°C.
ΔT = 660°C - 25°C = 635°C
Next, we need to convert the mass from kilograms to grams:
m = 3500 kg × 1000 g/kg = 3.5 x 10^6 g
Now we can calculate the heat:
Q = (3.5 x 10^6 g) * (0.897 J/g°C) * (635°C) = 1.98 x 10^9 J
The heat generated by the satellite falling to Earth is approximately 1.98 x 10^9 J.
Since this heat value is less than the change in mechanical energy (5.54 x 10^9 J), it is not sufficient to reach the melting point of aluminum.