Panic escape. The figure below shows a general situation in which a stream of people attempt to escape through an exit door that turns to be locked. The people move toward the door at speed vs = 3.7 m/s, are each d = 0.20 m in depth, and are separated by L = 1.75 m. The arrangement in the figure occuurs at time t = 0.

(a) At what average rate the layer of people at the door increase?
m/s

(b) At what time does the layer's depth reach 5.9 m? (The answer reveals how quickly such a situation becomes dangerous.)
s

A person covers the 1.75 m to the next person (stopped) in about 1.75 m/3.7 m/s = .473 seconds

so the stopping rate = 1/.473 = 2.11 people stop per second
that means the pile grows by
2.11*.2 = .423 m/second

then .423 m/s * t = 5.9 m
t = 13.9 seconds

To find the average rate at which the layer of people at the door increases, we need to determine the rate at which people reach the door and enter the layer.

(a) The rate at which people reach the door is equal to the speed at which they are moving, vs = 3.7 m/s.

To calculate the rate at which people enter the layer, we need to determine the time it takes for each person to reach the door.

Since people are separated by a distance of L = 1.75 m and they are moving at a speed of vs = 3.7 m/s, it takes each person t = L / vs = 1.75 m / 3.7 m/s = 0.47297 seconds to reach the door.

Therefore, the average rate at which the layer of people at the door increases is the reciprocal of the time it takes for each person to reach the door.

Average Rate = 1 / t = 1 / 0.47297 s ≈ 2.1147 s^-1

So, the average rate at which the layer of people at the door increases is approximately 2.1147 m/s.

(b) To find the time at which the layer's depth reaches 5.9 m, we need to calculate how many people have entered the layer.

Since each person has a depth of d = 0.20 m, the number of people required to create a layer with a depth of 5.9 m is n = 5.9 m / 0.20 m = 29.5 people.

Since the number of people must be a whole number, we can round up to the nearest whole number.

Therefore, it takes 30 people entering the layer for it to reach a depth of 5.9 m.

Since the rate at which people enter the layer is approximately 2.1147 s^-1 (calculated in part a), we can now determine the time it takes for 30 people to enter the layer.

Time = 30 people / 2.1147 s^-1 ≈ 14.18 seconds

So, it takes approximately 14.18 seconds for the layer's depth to reach 5.9 m.

To determine the average rate at which the layer of people at the door increases, we need to calculate the rate at which people are reaching the door.

(a) Calculate the time it takes for each person to reach the door:
Time = Distance / Speed
Time = d / vs = 0.20 m / 3.7 m/s ≈ 0.0541 s

Now, calculate the number of people that reach the door per unit time:
Number of people per unit time = 1 / Time = 1 / 0.0541 s ≈ 18.45 people/s

Since the people are separated by a distance L = 1.75 m, the average rate at which the layer of people at the door increases is:
Average rate = Number of people per unit time * Separation distance
Average rate = 18.45 people/s * 1.75 m = 32.26 m/s

Therefore, the average rate at which the layer of people at the door increases is approximately 32.26 m/s.

(b) To find the time at which the layer's depth reaches 5.9 m, we can use the relationship between the layer's depth, the separation distance, and the number of people that reach the door per unit time:
Layer's depth = Number of people per unit time * Separation distance * Time

Solving for time:
Time = Layer's depth / (Number of people per unit time * Separation distance)
Time = 5.9 m / (18.45 people/s * 1.75 m) ≈ 0.180 s

Therefore, the layer's depth reaches 5.9 m at approximately 0.180 s.