Figure 2-24 shows a general situation in which a stream of...

Figure $2-24$ 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 $v_{j}=3.50 \mathrm{~m} / \mathrm{s},$ are each $d=0.25 \mathrm{~m}$ in depth, and are separated by $I_{-}=1.75 \mathrm{~m} .$ The arrangement in Fig. $2-24$ occurs at time $t=0 .$ (a) At what average rate does the layer of people at the door increase? (b) At what time does the layer's depth reach $5.0 \mathrm{~m} ?$ (The answers reveal how quickly such a situation becomes dangerous.)

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Spatial Density and Distribution

The spatial density of a crowd is defined by the size of individuals and the distance between them. Understanding this distribution is crucial in scenarios involving crowd dynamics, as it influences how quickly a group accumulates at a narrow exit and how dangerous the situation can become.

Kinematics Relationships

Kinematics provides the framework to relate speed, distance, and time. By applying these relationships, one can calculate not only how fast individuals approach the door but also determine the time at which specific spatial thresholds, such as the increasing depth of the crowd, are reached.

Uniform Linear Motion

This concept involves objects moving at a constant speed in a straight line. It allows one to predict the future position of objects based solely on their speed, making it possible to calculate how fast a group of people moves towards an exit and how their positions change over time.

Rate of Change

Rate of change is a fundamental idea in calculus and physics that measures how one quantity alters with respect to another—typically time. In this context, it is used to determine how quickly the density or the depth of the crowd at the door increases, linking the movement of individuals with the changing accumulation of people at a fixed location.

Transcript

00:01 In this problem of motion along a straight line, we have given a figure i'm showing you.

00:07 So this is the figure.

00:08 Figure shows a general situation in which a stream of people attempt to escape through an exit door.

00:16 So this is the exit door.

00:19 That turns out to be locked.

00:21 So now this is in locked here.

00:24 The people move towards the door at a speed of vs 3 point, say people moved towards towards the door width vs this is 3 .5 meters per second equals to vs are each d is equal to 0 .5 meter in depth that means the depth of people is equal to say d is equal to 0 .25 meters and are at a separation of l is equal to 1 .75 meters and the separation between people is equal to 1 .75 meters and the separation between people is equals to 1 .75 meters or here we say that l is equal to 1 .75 meters.

01:16 The arrangement is here and it occurs at time t is equal to 0.

01:24 First at what average rate does the layer of people at the door increase.

01:31 So here we have to calculate the rate or we can say the average rate at which the layer of people at the door increases.

01:43 So first we would find the time taken by each people to reach the door.

01:50 Say this is the first people reaches the door.

01:53 The separation is here d is equal to 1 .75 meter and say vs is equal to 3 .5 meter per second...

Please give Ace some feedback