A 200m dash along a straight track was timed at 21.1 s by a timer located at the finish line who used the flash from the starters pistol to start the stopwatch. If the air temperature was 30.0c , what would the time have been if the timer had started the watch upon hearing the sound of the gun.
The speed of sound at that air temperature is about 345 m/s. The sound would not have arrived at the timer until 200/345 = 0.6s after the start of the race.
The recorded time would have been 20.5 s.
The delay due to the finite speed of light is negligible (about 1 microsecond).
yes
Well, if the timer had started the watch upon hearing the sound of the gun, I'm afraid it might have taken them a while to realize what happened. They'd probably be looking around confused, wondering where that sound came from, while all the runners have already crossed the finish line and gone for a post-race snack. So, in short, the time would have been somewhere between "late for the party" and "missed the bus."
To calculate the time difference between starting the stopwatch with the flash from the starters pistol and starting it upon hearing the sound of the gun, we need to consider the speed of sound and its impact on the timing.
1. Find the speed of sound: The speed of sound depends on the air temperature. We can use the approximate formula:
speed of sound = 331.4 + (0.6 * temperature in °C)
Plugging in 30.0 °C for the temperature:
speed of sound = 331.4 + (0.6 * 30)
speed of sound ≈ 331.4 + 18
speed of sound ≈ 349.4 m/s
2. Calculate the time difference: We need to determine how long it takes for the sound of the gun to reach the timer at the finish line. This time is equal to the distance traveled divided by the speed of sound.
distance = 200 m
time difference = distance / speed of sound
time difference = 200 / 349.4
time difference ≈ 0.572 s
3. Calculate the adjusted time: To find the adjusted time if the timer had started the watch upon hearing the sound of the gun, subtract the time difference from the original time.
adjusted time = original time - time difference
adjusted time = 21.1 - 0.572
adjusted time ≈ 20.528 s
Therefore, if the timer had started the watch upon hearing the sound of the gun, the time would have been approximately 20.528 seconds.
To solve this problem, we need to consider how temperature affects the speed of sound.
The speed of sound is directly proportional to the square root of the absolute temperature (Kelvin) of the medium through which it travels. The formula to calculate the speed of sound in air is as follows:
v = √(γ * R * T)
where:
v is the speed of sound in meters per second (m/s),
γ is the adiabatic index (approximately 1.4 for air),
R is the gas constant for air (approximately 287 J/(kg*K)),
T is the absolute temperature of the air in Kelvin (K).
Now, let's calculate the speed of sound at the given air temperature of 30.0°C (which we need to convert to Kelvin).
Step 1: Convert temperature from Celsius to Kelvin.
T(Kelvin) = T(Celsius) + 273.15
T(kelvin) = 30.0 + 273.15 = 303.15 K
Step 2: Plug the temperature into the speed of sound formula and calculate it.
v = √(γ * R * T)
v = √(1.4 * 287 * 303.15)
v ≈ 340.39 m/s
The speed of sound at 30.0°C is approximately 340.39 m/s.
Now, let's calculate the time it would have taken for the sound of the gun to reach the timer at the finish line.
The distance of the 200m dash is given as 200m, and we want to find the time it takes for the sound to travel this distance.
We can use the formula:
Time = Distance / Speed
Time = 200m / 340.39 m/s
Time ≈ 0.586 s
Therefore, if the timer had started the stopwatch upon hearing the sound of the gun, the time would have been approximately 0.586 seconds.
Please note that this calculation assumes ideal conditions and neglects other factors such as wind or atmospheric conditions that could affect the sound propagation.