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)? i got the wave speed right but cant get the value of width correct
To find the width of the wave, we need to determine the number of seats covered by the wave.
Given:
Distance traveled by the wave = 859 seats
Time taken for the wave to travel = 40.0 s
Time taken for spectators to respond = 1.80 s
To find the width of the wave, we can use the formula:
Width (w) = Wave speed (v) * Total response time (T)
(a) First, let's calculate the wave speed "v":
Wave speed (v) = Distance traveled / Time taken
Substituting the given values:
v = 859 seats / 40.0 s
v ≈ 21.48 seats/s
So, the wave speed is approximately 21.48 seats per second.
(b) To find the width of the wave (w), we need to calculate the total response time (T):
Total response time (T) = Time taken for spectators to respond * Number of seat waves
Number of seat waves = Distance traveled / Width of the wave
Substituting the given values:
Number of seat waves = 859 seats / w
Total response time (T) = 1.80 s * (859 seats / w)
Since the width (w) is equal to the distance traveled by the wave divided by the number of seat waves, we can substitute it into the equation:
Total response time (T) = 1.80 s * (859 seats / (859 seats / w))
= 1.80 s * w
Given that the total response time is 40.0 s, we can set up an equation:
40.0 s = 1.80 s * w
Simplifying the equation:
w = 40.0 s / 1.80 s
w ≈ 22.22 seats
So, the width of the wave is approximately 22.22 seats.
In summary, the wave speed is 21.48 seats per second, and the width of the wave is approximately 22.22 seats.