Question

At a baseball game, a spectator is \(60.0 \mathrm{m}\) away from the batter. How long does it take the sound of the bat connecting with the ball to travel to the spectator's ears? The air temperature is \(27.0^{\circ} \mathrm{C}\)

Short answer

Answer: Approximately 0.1725 seconds.

Step-by-step solution

Find the speed of sound in air at the given temperature

To find the speed of sound at the given temperature, we can use the formula: \(v = 331.4 + 0.6T\) where \(v\) is the speed of sound in meters per second and \(T\) is the temperature in degrees Celsius.

Substitute the temperature value into the formula

In this case, the air temperature is \(27.0^{\circ} \mathrm{C}\). So, substitute the value of \(T\) in the formula: \(v = 331.4 + 0.6(27)\).

Calculate the speed of sound

Calculate the right-hand side of the equation: \(v = 331.4 + 0.6(27) = 331.4 + 16.2 = 347.6 \mathrm{m/s}\). The speed of sound in air at the given temperature is \(347.6\mathrm{m/s}\).

Use the distance and speed of sound to find the time

The spectator is \(60.0 \mathrm{m}\) away from the batter, and the speed of sound in air at the given temperature is \(347.6 \mathrm{m/s}\). To find the time it takes for the sound to travel this distance, we can use the formula: \(t = \frac{d}{v}\), where \(t\) is the time, \(d\) is the distance, and \(v\) is the speed.

Substitute the distance and speed values into the formula

Substitute the values of \(d\) and \(v\) in the formula: \(t = \frac{60.0}{347.6}\).

Calculate the time it takes for the sound to reach the spectator

Calculate the right-hand side of the equation: \(t = \frac{60.0}{347.6} \approx 0.1725 \mathrm{s}\). It takes approximately \(0.1725 \mathrm{s}\) for the sound of the bat connecting with the ball to reach the spectator's ears at the given conditions.