A steel ball is dropped from a building's roof and passes a window, taking 0.125 s to fall from

step 1 Okay, we've got a ball being dropped from a roof. It takes this much time to drop from the tops

A steel ball is dropped from a building's roof and passes a...

Key Concepts

Uniformly Accelerated Motion

This concept deals with the movement of objects under a constant acceleration, such as gravity. It provides the basis for analyzing the motion of a falling object by using a fixed acceleration value (g = 9.8 m/s²) to calculate velocities, distances, and times during free-fall.

Kinematic Equations

These equations relate displacement, initial velocity, final velocity, acceleration, and time for objects moving under constant acceleration. They are fundamental tools for solving problems involving free falling bodies and other projectile motions, allowing one to determine unknown parameters when certain initial conditions are known.

Projectile Motion Symmetry

In projectile motion, particularly when air resistance is negligible, the time taken for the object to travel upwards past a given point equals the time to travel downward past that same point. This symmetry simplifies calculations and is used here when assuming that the upward return of the ball mirrors the downward passage.

Time and Distance Analysis

The relationship between time intervals and distances traveled forms an essential part of solving motion problems. By breaking the motion into segments (for example, time spent in free fall and time spent rising after a bounce), one can set up equations to calculate unknown distances or the heights of objects, as demonstrated in the analysis of the ball's passage relative to the window.

Transcript

00:01 Okay, we've got a ball being dropped from a roof.

00:08 It takes this much time to drop from the tops of the bottom of the window, which is a distance of 1 .2 -0 meters, okay? but then it bounces upward and does the same thing in the same amount of time.

00:48 Okay? interesting.

00:51 The flight upward is the exact reverse of the fall.

01:06 Okay, so we're assuming a perfectly elastic collision.

01:12 So the bounce up is exactly the same as the bounce as going down.

01:21 So that means if it spends two seconds below the window, that means that one second of it was falling and the other second was rising.

01:31 So i'm going to write t -sou.

01:35 Below is one second falling.

01:40 Okay.

01:42 How tall is the building? well, the initial is zero on the top of the building.

01:51 Okay.

01:53 So, um, interesting.

02:10 Okay.

02:12 Well, we know why.

02:18 From the top to the bottom, and we don't know the height of the ball, but on the bottom, it's zero.

02:28 So i guess i'll just draw top to bottom here.

02:35 Y equals y initial plus v initial t, which is also zero, minus one -half g t squared.

02:58 Okay.

02:59 Okay.

03:02 Now, why to the, we've got to do something with the window.

03:27 We don't know the height of the window.

03:32 I mean, well, we don't know the height of the window.

03:36 But let's do top of the window now.

03:54 So, why, whoops, why to the top of the window, which i'm just going to call w equals y initial, which is going to equal the height of the building, minus, because v0 is zero, one -half g, t to the top of the window squared.

04:52 Okay? well, let's do bottom of the window.

05:09 But we know t to the bottom of the window minus t.

05:16 To the top of the window is going to equal t we also know the distance so we also know that y sub w minus y sub b is d so let's see i don't know t b wait a minute i just wrote t b twice okay, that's not the same t sub b.

06:05 So, t sub, i guess i'm going to write here.

06:20 I'm just going to call that t sub 2.

06:25 Okay.

06:27 So, i know i have four equations, and, i mean, technically i have five equations, but i don't know h, i don't know t, i don't know t, i don't know tw or tb, and i don't know yw or yb, but i know everything else.

07:14 Don't know the height of the building.

07:17 I don't know total time, yes i do, because i have t equals 0 .125.

07:32 Oh, no, that's a different t.

07:35 Oh, man.

07:48 So this is a different t up here.

07:51 This is the total time.

08:04 And i have not yet used this.

08:10 So anyway, i don't know the total time.

08:18 I don't know h.

08:24 I don't know time to the window.

08:29 I don't know time to the bottom of the window.

08:31 I don't know height of the window.

08:32 I don't know that.

08:35 Okay.

08:37 So six unknown.

08:38 And five equations.

08:40 But i do know that the time below the window is one second.

08:52 So i do know that the time below the window.

09:15 Okay.

09:24 So if i take the time to reach the window plus the time at the window, plus the time below the window, then that's going to give me the total time.

10:07 So now i think i have six equations and six unknowns.

10:14 And so i should be able to solve it.

10:17 Time to the window plus time being seen in the window, plus the time below the window would be the total time.

10:30 Okay.

10:32 So, what am i trying to solve for? it's trying to solve for h.

10:42 So, up at the top, 1 .5g tt squared equals h.

11:00 So let's solve this for t squared, or just t -t.

11:15 2h over g, and then we take the square root of that.

11:24 So anywhere i see t -sub -t, t -sub -t, i can write square root of 2h over g.

11:39 Now, down to five equations, because i took t sub t out of it.

11:46 Okay.

11:49 Y equals, i'm just double checking here.

11:51 Why equals y initial minus one half, minus one half, tb minus tw equals t.

12:01 Yes.

12:03 Y, w minus y, b is d.

12:05 Yes.

12:06 Okay.

12:09 So i think i'm good here.

12:12 Okay, so now i've got to look at these five equations, and i'm trying to solve for h.

12:19 So i can take t out of the equation by substituting this into there.

12:33 So now i've got t sub w plus t sub b minus t sub w.

12:46 Plus t sub two equals square root of two h, whoops, over g.

12:54 So i can take t sub w, or that cancels out t sub w, but i've also, the main thing i did was i removed t from the equation.

13:06 So i've got t sub b plus t sub two is the square root of 2h over g.

13:15 Okay.

13:16 So now i've got tb, t2, h, tb, oh, no, i know t2, t2, tb and t2, tb and tb and yw.

13:45 Okay.

13:51 Well, if i take this and substitute it in here, i can get rid of yw.

13:57 Yw.

13:59 Yw is yb plus d, h minus 1⁄2gtw squared.

14:23 So i've got that equation, and let me just write down the third equation, yb equals h minus one half gtb squared.

14:40 Okay, so now i'm down to three equations and three unknowns.

14:44 I'm just making sure i didn't write anything incorrectly here.

14:55 And it looks good.

14:57 So now i'm down to these three equations.

15:00 I need to know tb, uh -oh, tw, tw, t -b, t -w, y, b, and h.

15:41 Okay, so let me think here.

15:46 Okay, i've got four unknowns here.

15:49 I've got to get t .w out of there.

15:54 But i know that tw is tb minus t.

16:00 So, but wait a minute.

16:18 I don't know t.

16:20 Yes, i do.

16:22 Yes, i do.

16:29 So now i'm confused a little bit because right here, i wrote tb minus tw equals t.

16:53 And i replaced t with tb minus tw.

16:58 Well, that was crazy because i know t.

17:01 All right, so that was a weird way to proceed.

17:05 But nevertheless, it does get us down to here.

17:09 Take this one out.

17:11 So i made a little mistake in that i substituted and removed t from the equation, but i made up for it down here.

17:20 Okay, so now i need to know tb.

17:23 I need to know h, and i need to know yb.

17:30 But i know everything else.

17:37 Okay.

17:40 So, i see tb here.

17:48 I see tb here, and i see tb here...

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