On a beautiful baseball day with air temperature of 200 C, a pitcher throws a 0.142 kg baseball at 51 m/s towards home plate. As it travels 19.4 m, the ball slows down to a speed of 36 m/s because of air resistance. Calculate the change in the temperature of the air through which the ball passes. For air M= 28.9 g/mol and Cp =7R/2. Hint: Assume the change in temperature happens only for a cylinder of air 19.4 m in length and radius of 3.7 cm. Assume that the energy loss of the ball due to its change in KE and is all absorbed by the air
To calculate the change in temperature of the air through which the ball passes, we need to determine the energy absorbed by the air due to the change in kinetic energy of the ball.
Step 1: Calculate the initial kinetic energy (KE₁) of the ball.
The formula for kinetic energy is KE = (1/2)mv², where m is the mass and v is the velocity.
Given:
- Mass of the baseball (m) = 0.142 kg
- Initial velocity (v) = 51 m/s
Using the formula, we can calculate the initial kinetic energy:
KE₁ = (1/2) * 0.142 kg * (51 m/s)²
Step 2: Calculate the final kinetic energy (KE₂) of the ball.
Given:
- Final velocity (v) = 36 m/s
Using the same formula, we can calculate the final kinetic energy:
KE₂ = (1/2) * 0.142 kg * (36 m/s)²
Step 3: Calculate the change in kinetic energy (ΔKE) of the ball.
The change in kinetic energy is the difference between the initial and final kinetic energies:
ΔKE = KE₁ - KE₂
Step 4: Calculate the energy absorbed by the air.
Since the energy loss of the ball is all absorbed by the air, the energy absorbed by the air is equal to the change in kinetic energy of the ball:
Energy absorbed by air = |ΔKE|
Step 5: Calculate the temperature change (ΔT) of the air.
The energy absorbed by the air can be determined using the equation:
Energy absorbed by air = n * Cp * ΔT
Where:
- n is the number of moles of air in the cylinder
- Cp is the specific heat capacity of air (which is given as 7R/2)
- ΔT is the change in temperature of the air
To rewrite the equation, we can rearrange it as:
ΔT = (Energy absorbed by air) / (n * Cp)
Now, let's calculate the temperature change:
- Convert the mass of air to moles:
Given:
- Mass of air (M) = 28.9 g/mol
n (number of moles of air) = (Mass of air) / (molar mass of air)
Substituting the given values:
n = (28.9 g) / (28.9 g/mol)
- Calculate the temperature change ΔT:
ΔT = (Energy absorbed by air) / (n * Cp)
Finally, substitute the value of the energy absorbed by air and solve for ΔT.
To calculate the change in temperature of the air through which the ball passes, we need to determine the heat transferred from the baseball to the air as it slows down.
First, let's calculate the initial and final kinetic energies of the baseball:
Initial kinetic energy (KE1) = (1/2) * mass * velocity^2
= (1/2) * 0.142 kg * (51 m/s)^2
Final kinetic energy (KE2) = (1/2) * mass * velocity^2
= (1/2) * 0.142 kg * (36 m/s)^2
The change in kinetic energy of the baseball is given by the difference between these two values:
ΔKE = KE2 - KE1
Now, the heat transferred from the baseball to the air can be calculated using the formula:
ΔQ = ΔKE
Since the energy loss of the ball due to its change in kinetic energy is all absorbed by the air, the heat transferred to the air is equal to the change in kinetic energy of the baseball.
Next, we need to determine the number of moles of air in the cylinder through which the ball passes. We can use the ideal gas equation (PV = nRT) to calculate the number of moles.
The volume of the cylinder is given by:
V = π * r^2 * h
= π * (0.037 m)^2 * 19.4 m
Where r is the radius of the cylinder (converted from 3.7 cm to meters), and h is the height of the cylinder, which is given as 19.4 m.
Now, we substitute the values into the ideal gas equation:
PV = nRT
Where P is the pressure of the air, V is the volume, n is the number of moles of air, R is the ideal gas constant, and T is the temperature.
We are trying to find the change in temperature (ΔT), so we rearrange the equation as:
ΔT = ΔQ / (n * Cp)
Where Cp is the specific heat capacity at constant pressure for air, given as 7R/2.
Now, we have all the values needed to calculate the change in temperature of the air. Plug in the appropriate values into the equation:
ΔT = ΔQ / (n * Cp)
Finally, calculate the change in temperature using the calculated values.