On a beautiful baseball day with air temperature of 200 C, a pitcher throws a 0.142 kg baseball at 49 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.
200 C?
200 degrees Celcius
Huh? The boiling point of water is 100 degrees Celsius.
To calculate the change in temperature of the air through which the ball passes, we need to determine the energy loss of the ball due to its change in kinetic energy and then equate it to the energy gained by the air. Let's break down the steps to get to the answer:
Step 1: Calculate the initial kinetic energy of the ball
The initial kinetic energy (KE_initial) of the ball can be determined using the formula:
KE_initial = (1/2) * m * v_initial^2
Here, m is the mass of the ball (0.142 kg) and v_initial is the initial velocity (49 m/s). Substitute the values in and calculate KE_initial.
KE_initial = (1/2) * 0.142 kg * (49 m/s)^2
Step 2: Calculate the final kinetic energy of the ball
The final kinetic energy (KE_final) of the ball can be determined in the same manner. The mass and final velocity values are the same as before, only the velocity changes.
KE_final = (1/2) * m * v_final^2
Here, v_final is the final velocity (36 m/s). Substitute the values in and calculate KE_final.
KE_final = (1/2) * 0.142 kg * (36 m/s)^2
Step 3: Calculate the change in kinetic energy of the ball
The change in kinetic energy (ΔKE) of the ball is the difference between the initial and final kinetic energies.
ΔKE = KE_initial - KE_final
Calculate ΔKE using the previously determined values. Note that this value represents the energy loss of the ball.
Step 4: Calculate the energy gained by the air
The energy gained by the air can be calculated using the formula:
Energy_gained = n * Cv * ΔT
Here, n is the number of moles of air, Cv is the specific heat capacity of the air at constant volume, and ΔT is the change in temperature of the air.
We are given the molar mass of air (M = 28.9 g/mol), and Cp = 7R/2. Since Cv = Cp - R, we can calculate Cv. The value of R is 8.314 J/(mol·K).
Cv = Cp - R = (7R/2) - R = (5R/2)
Step 5: Calculate the number of moles of air
The number of moles of air (n) can be calculated using the formula:
n = mass / M
Here, mass is the mass of the cylinder of air through which the ball passes. The mass can be calculated by multiplying the density of air by the volume of the cylinder.
mass = density * volume
Density of air = M * pressure / (R * temperature)
Temperature = 200 °C = 200 + 273.15 K (converting to Kelvin)
Volume = π * (radius)^2 * length
Substitute the given values into the above equations to calculate the mass and subsequently the number of moles.
Step 6: Calculate the change in temperature of the air
Now that we have the energy loss of the ball (ΔKE) and the energy gained by the air (Energy_gained), we can equate them to find the change in temperature of the air (ΔT).
ΔKE = Energy_gained
Substitute the previously calculated values into the equation and solve for ΔT.
After completing these steps, you would have calculated the change in the temperature of the air through which the ball passes.