A ball of volume V=8 L is full of air with pressure higher than atmospheric pressure by Δp=2×10^4 Pa. The total mass of the ball and the air inside is 200 g. The ball is tossed up to a height h=20 m, then it falls down, collides with the ground, and bounces back up. Estimate the highest temperature the air inside the ball could reach during the collision in Celsius.
Details and assumptions
The gravitational acceleration is g=−9.8 m/s^2.
The atmospheric pressure is po=10^5 Pa.
The specific heat of air is C=700 J/kgK.
The temperature of the surrounding environment is To=27∘C.
The mass per mol of air is μ=29 g/mol.
The gas constant is R=8.31 J/molK.
Treat air as an ideal gas to simplify the problem.
Neglect air resistance.
16.5
that's correct 16.5
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87.45
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To estimate the highest temperature the air inside the ball could reach during the collision, we can use the principle of conservation of energy, taking into account the changes in potential energy and internal energy.
1. Determine the initial and final states of the air inside the ball:
The initial state is when the ball is at ground level and the air pressure inside is higher than atmospheric pressure. The final state is when the ball reaches its highest point during the bounce and momentarily comes to rest.
2. Calculate the change in potential energy:
The change in potential energy is given by:
ΔPE = mgh
where m is the total mass of the ball and air inside, g is the gravitational acceleration, and h is the height reached by the ball. In this case, m = 200 g = 0.2 kg, g = -9.8 m/s^2, and h = 20 m. Plug in these values to calculate ΔPE.
3. Calculate the change in internal energy:
The change in internal energy is given by:
ΔU = ΔQ - ΔW
where ΔQ is the heat added to the system and ΔW is the work done by the system. Since we are neglecting air resistance, ΔW is zero. Hence, ΔU = ΔQ.
4. Determine the heat added to the system:
The heat added to the system can be calculated using the equation:
ΔQ = nCΔT
where n is the number of moles of air, C is the specific heat of air, and ΔT is the change in temperature. To calculate n, divide the mass of the air (200 g) by the molar mass of air (μ = 29 g/mol). Now you can calculate ΔQ.
5. Equate the change in potential energy and the change in internal energy:
ΔPE = ΔU
This is based on the principle of conservation of energy. Substitute the values calculated for ΔPE and ΔQ to get the equation.
6. Solve the equation for ΔT:
Rearrange the equation to solve for ΔT:
ΔT = ΔPE / (nC)
Substitute the values calculated for ΔPE, n, and C to get the value of ΔT.
7. Convert ΔT from Kelvin to Celsius:
To convert ΔT from Kelvin (K) to Celsius (°C), subtract 273.15 from the value of ΔT.
The resulting ΔT in Celsius is the estimated highest temperature the air inside the ball could reach during the collision.