Position, Displacement, and Average Velocity Panic escape....

Position, Displacement, and Average Velocity Panic escape. 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 $v_{s}=3.50 \mathrm{m} / \mathrm{s},$ are each $d=0.25 \mathrm{m}$ in depth, and are separated by $L=1.75 \mathrm{m} .$ 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.)

Position, Displacement, and Average Velocity
Panic escape. 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 $v_{s}=3.50 \mathrm{m} / \mathrm{s},$ are each $d=0.25 \mathrm{m}$ in depth, and are separated by $L=1.75 \mathrm{m} .$ 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.)

Key Concepts

Position

Position is a fundamental kinematics concept that defines the location of an object relative to a chosen reference point or coordinate system. It is typically specified as a vector, which includes both magnitude and direction, to indicate where an object is at a given time.

Displacement

Displacement is the net change in position of an object, calculated as the difference between its final and initial positions. Unlike distance, which is a scalar and always positive, displacement is a vector quantity that takes direction into account, making it crucial for understanding how far and in what direction an object has moved.

Average Velocity

Average velocity is defined as the total displacement divided by the total time taken for that displacement. It provides a measure of the overall rate of change of position over a given time interval and is distinct from the instantaneous velocity, which can vary moment by moment.

Rate of Change

Rate of change is a general concept in both physics and mathematics that quantifies how one variable changes with respect to another, such as time. In kinematics, it is used to analyze how quantities like position change over time, providing insight into the dynamics of a moving object.

Transcript

00:01 In this problem, we have a group of people, and i'll just draw it out to illustrate things.

00:14 And so there's space at a distance l apart.

00:20 People all have a depth of d.

00:23 So this is not to scale.

00:30 And we know the values of these things.

00:32 So l is 1 .75 meters, and d is 0 .25 meters.

00:47 Okay, and we're also told that people are all moving up.

00:50 A speed equal to 3 .5 meters per second.

01:00 And so what's happening is this sort of line of people is approaching this door.

01:06 And once they run into the door, they slam into it, they get stuck.

01:12 And so if we think about a, what's going on here? the people move towards the door, they get stuck and we get this layer of people that build up and increases in thickness at the door.

01:26 And so a is asking us to calculate the rate of increase of the layer of the people at the door.

01:33 Let's just think about what's meant by a rate here.

01:39 So we're increasing by some unit size for a given amount of time.

01:48 Right, so we know that time is going to be in the denominator and some sort of size parameter.

01:57 It's going to have to go in the numerator.

01:59 And so let's just think about this for a second.

02:03 So the units that we're adding to the layer are going to be the thickness of the people, d.

02:11 So if we plug in d for the numerator, we're saying that the rate of increase of thickness of this layer is equal to units of d per time.

02:22 And so that's how we're going to want to write this out.

02:27 And so again, we're given the value of d, we know that the value of the value.

02:31 0 .25 meters, but we're not given the value of t.

02:35 So let's write out the definition of speed, but i'm going to be very careful to write it out in terms of the context of this problem.

02:45 So when we have a speed, we know that it's a distance per time.

02:52 So we're going to throw a time in the denominator.

02:54 And i just want to think about, we just want to think about the relevant distance to put in the numerator here...

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