A small piece of Styrofoam packing material is dropped from...

A small piece of Styrofoam packing material is dropped from a height of $2.00 \mathrm{m}$ above the ground. Until it reaches terminal speed, the magnitude of its acceleration is given by $a=g-B v$. After falling $0.500 \mathrm{m},$ the Styrofoam effectively reaches terminal speed and then takes 5.00 s more to reach the ground. (a) What is the value of the constant $B$ ? (b) What is the acceleration at $t=0 ?(\mathrm{c})$ What is the acceleration when the speed is $0.150 \mathrm{m} / \mathrm{s}$ ?

Key Concepts

Free Fall and Kinematics

Free fall motion can be described using kinematics, where the acceleration due to gravity is initially the sole force acting on the object. When air resistance (drag) is taken into account, the acceleration becomes a function of velocity, as given by a = g - Bv. This relationship modifies the standard kinematics by introducing a velocity-dependent term, which is essential for analyzing the motion both before reaching terminal velocity and after a constant velocity phase is established.

Linear Drag Force

A linear drag force is one that is directly proportional to the velocity of the object, often applicable in situations involving low-speed motion through a fluid. This means the resistive (drag) force can be written as F_drag = Bv, where B is a constant. In the context of free fall, this drag force reduces the net acceleration of the object over time until terminal velocity is reached.

Terminal Velocity

Terminal velocity is the constant speed achieved by an object when the net force acting on it becomes zero. In problems involving air resistance where the drag is proportional to velocity, the object reaches terminal velocity when the gravitational force is exactly balanced by the drag force. This condition is mathematically represented by setting the acceleration (g - Bv) equal to zero, which gives v_terminal = g/B.

Transcript

00:01 Okay, so in this problem, we have styrofoam, dropped from a height of 2 .00 meters.

00:08 Let's write that down.

00:13 It reaches terminal speed.

00:16 The acceleration is g minus bv, where b is some constant, and then acceleration is proportional to the velocity.

00:28 So after following the half meter, it reaches its terminal speed.

00:33 So let's just draw this out.

00:35 So here's the styro from, and then it reaches vt, and then it's, let's just call this, i don't know, what should i call this? i'll call this ht, i guess, to associate it with the height that it falls before it reaches its terminal velocity, and then that's 0 .5 .0 meters.

01:01 And then to go from this point to the ground, it takes five seconds.

01:08 So i'll call that t.

01:12 Then we want to get the value of the constant b.

01:18 Okay.

01:20 So lots of little pieces to the puzzle.

01:25 And we want the constant b and then the acceleration when t is equal to zero.

01:34 So i instead of like focusing on what we want, i'm just going to figure out like what we could find using this information.

01:42 So we know that this whole height is to.

01:47 And then this is a half equals .5.

01:55 Oh, i think i'm gonna work down here.

01:58 I think that's where we have the most information because we know it falls two minus point five is 1 .5 meters.

02:06 So this whole section is 1 .5 meters.

02:12 And we know that it's our constant velocity of whatever the terminal velocity is.

02:18 So we know we can say vt, is equal to 1 .5 meters divided by five seconds.

02:31 So that's 3 over 10.

02:33 So that's 0 .3 meters per second.

02:43 So that's its terminal velocity.

02:46 And then, okay...

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