A human wave. During sporting events within large, densely packed stadiums, spectators will send a wave (or pulse) around the stadium (Fig. below). As the wave reaches a group of spectators, they stand with a cheer and then sit. At any instant, the width w of the wave is the distance from the leading edge (people are just about to stand) to the trailing edge (people have just sat down). Suppose a human wave travels a distance of 859 seats around a stadium in 40.0 s, with spectators requiring about 1.80 s to respond to the wave's passage by standing and then sitting. What are (a) the wave speed v (in seats per second) and (b) width w (in number of seats)?
To find the wave speed and width, we need to use the given information and formulas related to speed, distance, and time.
a) The wave speed (v) can be calculated using the formula:
v = distance / time
Given that the distance traveled by the wave is 859 seats and the time taken is 40.0 seconds, we can substitute these values into the formula:
v = 859 seats / 40.0 s
Calculating the value:
v = 21.475 seats/s
Therefore, the wave speed is approximately 21.475 seats per second.
b) The width of the wave (w) can be calculated using the formula:
w = wave speed * response time
Given that the wave speed is 21.475 seats/s and the response time is 1.80 s, we can substitute these values into the formula:
w = 21.475 seats/s * 1.80 s
Calculating the value:
w = 38.655 seats
Therefore, the width of the wave is approximately 38.655 seats.