The figure below shows a general situation in which a stream of people attempt to escape through an exit door that turns out to be locked. The people move toward the door at speed v = 4.1 m/s, are each d = 0.30 m in depth, and are separated by L = 1.75 m. The arrangement in the figure occurs at time t = 0.

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

(b) At what time does the layer's depth reach 4.1 m?

a person gets to the door each 1.75/4.1 seconds. So depth increases .3m each time.

rate=.3/(1.75/4.1) m/s

To answer these questions, we need to analyze the situation and understand the motion of the people.

(a) Average rate at which the layer of people at the door increases:

The layer of people at the door increases when new people reach the door. To calculate the average rate, we need to determine how many people reach the door per unit time.

The distance between each person is L = 1.75 m, and they are moving at a speed of v = 4.1 m/s. If we divide the distance between each person (L) by their speed (v), we can find the time it takes for one person to reach the door:

Time for one person = L / v = 1.75 m / 4.1 m/s ≈ 0.4268 s

So, on average, one person reaches the door every approximately 0.4268 seconds.

To find the average rate at which the layer of people at the door increases, we divide the depth of each person (d) by the time it takes for one person to reach the door:

Average rate = d / (L / v) = 0.30 m / 0.4268 s ≈ 0.7032 m/s

Therefore, the average rate at which the layer of people at the door increases is approximately 0.7032 m/s.

(b) Time at which the layer's depth reaches 4.1 m:

To calculate the time at which the layer's depth reaches 4.1 m, we need to divide the target depth by the rate at which the layer increases:

Time = depth / rate = 4.1 m / 0.7032 m/s ≈ 5.82 s

Therefore, the layer's depth will reach 4.1 m at approximately 5.82 seconds.

Note: It's important to note that this analysis assumes a constant speed and uniform distribution of people. In reality, the speed and distribution may vary, so these calculations provide an approximation.