A ball of volume V=8 L is full of air with pressure higher than atmospheric pressure by ?p=2×104 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/s2.
The atmospheric pressure is po=105 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.
31.86
To estimate the highest temperature the air inside the ball could reach during the collision, we can make use of the principles of thermodynamics and ideal gas laws. Here are the steps to calculate it:
Step 1: Determine the initial and final states of the air inside the ball during the collision.
- Initial state: The air inside the ball is at its initial temperature and pressure. We are given that the pressure inside the ball is higher than atmospheric pressure by Δp = 2×104 Pa.
- Final state: The air inside the ball will experience a sudden change in pressure and compression during the collision with the ground, resulting in an increase in temperature.
Step 2: Calculate the change in volume of the air inside the ball during the collision.
Since the ball is tossed up to a height of h = 20 m and then falls back down, the volume of the air inside the ball will change. We need to find the change in volume (ΔV) during this process.
Using the ideal gas law equation, PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the gas constant, and T is the temperature, we can write:
P1V1 = nRT1 --(equation 1) (Initial state)
P2V2 = nRT2 --(equation 2) (Final state)
Since the number of moles (n) remains constant throughout the process, we can divide equation 2 by equation 1:
P2V2 / P1V1 = (nRT2) / (nRT1)
V2 / V1 = T2 / T1 (canceling n, R, and rearranging the equation)
Step 3: Calculate the change in temperature of the air inside the ball during the collision.
Using equation 3, we can calculate the change in temperature (ΔT) during the collision:
ΔT = T2 - T1
Step 4: Calculate the highest temperature the air inside the ball could reach.
Since we are interested in the highest temperature, we need to find the maximum value of ΔT. This occurs when the volume change is the greatest.
To estimate this, we can assume that the ball behaves ideally, and that the pressure change Δp is constant throughout the collision. Therefore, the initial and final pressures can be written as:
P1 = po + Δp --(equation 4) (Initial pressure)
P2 = po --(equation 5) (Final pressure)
Here, po is the atmospheric pressure.
Step 5: Calculate the mass of air inside the ball.
Given the total mass of the ball and air inside is 200 g, we need to determine the mass of air (m) inside the ball.
Step 6: Calculate the change in volume during the collision.
Using the ideal gas law equation, we can rewrite equation 5 as:
P2V2 = mRT2/V2 --(equation 6)
Rearranging equation 6, we can solve for V2:
V2^2 = mR(T2/P2)
Substituting equation 4 into equation 6:
V2^2 = mR(T2/(po + Δp))
Since we are assuming the volume change is small, we can use the approximation that V2 ≈ V1 + ΔV:
(V1 + ΔV)^2 ≈ mR(T2/(po + Δp))
Expanding and neglecting higher-order terms:
V1^2 + 2V1ΔV ≈ mR(T2/(po + Δp))
Since the initial volume V1 is given as 8 L, we have:
64 + 16ΔV ≈ mR(T2/(po + Δp))
Dividing both sides by 16ΔV:
4 + ΔV/V1 ≈ mR(T2/(16Δp))
Rearranging the equation to express ΔV/V1 in terms of m, Δp, and T2:
ΔV/V1 ≈ (mR(T2/(16Δp))) - 4
Step 7: Calculate the change in temperature (ΔT) during the collision.
Using equation 3, we can relate the change in volume (ΔV/V1) and the change in temperature (ΔT):
V2 / V1 = T2 / T1
Substituting ΔV/V1 for V2/V1, we get:
T2 / T1 ≈ 1 + ΔT / T1
Expanding the equation and neglecting higher-order terms:
T2 ≈ T1 + ΔT
Rearranging the equation to express ΔT in terms of T1:
ΔT ≈ T2 - T1 ≈ (T1)(ΔV/V1)
Step 8: Convert the change in temperature to Celsius.
To convert the change in temperature ΔT from Kelvin to Celsius, subtract 273.15 from ΔT.
Step 9: Calculate the highest temperature the air inside the ball could reach.
The highest temperature occurs when the air inside the ball reaches its maximum compression. Therefore, to estimate the highest temperature (Tmax), we calculate ΔT for the maximum compression using equation 7.
Using the given values and equations, you can now follow the steps to estimate the highest temperature the air inside the ball could reach during the collision.