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.
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To estimate the highest temperature the air inside the ball could reach during the collision, we can use the ideal gas law and the principle of conservation of energy.
Step 1: Calculate the initial pressure of the air inside the ball:
Since the air inside the ball has a pressure higher than atmospheric pressure by Δp=2×10^4 Pa, we can calculate the initial pressure of the air:
Initial pressure = Atmospheric pressure + Δp = 1 atm + 2×10^4 Pa
Step 2: Calculate the final pressure of the air inside the ball:
When the ball is at the highest point of its trajectory (h=20m), the pressure inside the ball is equal to atmospheric pressure. Therefore, the final pressure of the air inside the ball is the atmospheric pressure: Final pressure = Atmospheric pressure = 1 atm
Step 3: Calculate the work done by the air during compression:
When the ball is on the way up, the air inside the ball is compressed. The work done by the air during compression is given by:
Work = (Final pressure - Initial pressure) * Change in volume
To find the change in volume, we need to calculate the difference in radii of the ball before and after compression. Assuming the ball is a perfect sphere, the change in volume is given by:
Change in volume = (4/3) * π * [(Initial radius)^3 - (Final radius)^3]
Step 4: Calculate the final temperature of the air inside the ball:
Using the principle of conservation of energy, the work done on the air during compression is equal to the increase in internal energy of the air, which can be expressed as:
Work = Change in internal energy = mass * specific heat capacity * (Final temperature - Initial temperature)
Since the ball and the air inside have a total mass of 200 g, we can use this mass value in the equation. The specific heat capacity of air is approximately 1004 J/kg°C.
Step 5: Calculate the highest temperature:
Now we can rearrange the equation from step 4 to solve for the final temperature:
Final temperature = (Work / (mass * specific heat capacity)) + Initial temperature
By substituting the values we calculated in the previous steps, including the initial temperature of the air inside the ball, we can estimate the highest temperature the air could reach during the collision.
Please note that this is an estimation, and there may be other factors to consider.