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8.01x - Lect 2 - 1D Kinematics - Speed, Velocity, Acceleration — Transcript

by Lectures by Walter Lewin. They will make you ♥ Physics. · 6,444 words · 858 segments · language en · Watch on YouTube

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0:01We will discuss velocities and acceleration.
0:06I'll start with something simple.
0:08I have a motion of an object along a straight line--
0:13we'll call that one-dimensional motion.
0:16And I'll tell you that the object is here at time t1.
0:19At time t2, it's here.
0:20At time t3, it's there.
0:22At time t4, it's here
0:24and at time t5, it's back where it was at t1.
0:27And here you see the positions in x
0:30where it is located at that moment in time.
0:38I will define this to be the increasing value of x.
0:43It's my free choice, but I've chosen this now.
0:47Now we will introduce what we call the average velocity.
0:53I put a bar over it.
0:55That stands for average between time t1 and time t2.
0:59That we define in physics
1:01as x at time t2 minus x at time t1
1:06divided by t2 minus t1.
1:10That is our definition.
1:12In our case, because of the way that I define
1:15the increasing value of x, this is larger than 0.
1:21However, if I take the average velocity between t1 and t5
1:28that would be 0, because they are at the same position
1:32so the upstairs is 0.
1:34If I had chosen t4 and t2--
1:39average velocity between time t2 and t4--
1:42you would have seen that that is negative
1:44because the upstairs is negative.
1:49Notice that I haven't told you
1:50where I choose my zero on my x axis.
1:54It's completely unimportant for the average velocity.
1:57It makes no difference.
1:59However, if I had chosen this
2:02to be the direction of increasing x
2:06then, of course, the signs would flip.
2:08Then this would have been negative
2:10and this would have been positive.
2:12So the direction, that you are free to choose
2:15determines the signs.
2:17The location where you put your zero is not important
2:20but signs in physics do matter.
2:23Signs are important.
2:25Whether you owe me money or I owe you money
2:29the difference is only a minus sign
2:31but I think it's important for you.
2:35Now I will give you not only the positions--
2:40as I did here on the x axis at discrete moments in time--
2:45but I'm going to tell you
2:47exactly where the object is at any moment in time.
2:50Here you see an xt diagram
2:53so you see that at t1, the object is at position xt1.
2:59This is the road of the object.
3:00This is the straight line, where it's moving.
3:02It starts here and it goes to this position.
3:06It goes to this one, it comes back to t4
3:07and it comes back here.
3:09I will tell you now every moment in time in between.
3:20There it goes.
3:25Voilà.
3:27This is now information that is way more.
3:30You have the information at any moment in time.
3:32Notice that I now did choose x = 0.
3:37I chose it somewhere here
3:39but I could have chosen it at any other point--
3:42for whatever follows
3:43you will see that it makes no difference--
3:45so I have chosen a zero point so that I can make a graph.
3:49And now we will look at the average velocity
3:53in a somewhat different way.
3:55Say I choose my time t2 and t3.
3:59I draw here now this line.
4:09And this angle I call alpha
4:12and this part here I call delta x
4:19and this here is delta t.
4:24And so you could right now--
4:26if you're careful about your sign convention--
4:28you could write down now that the average velocity
4:32equals delta x divided by delta t.
4:35But be careful.
4:37If the angle is positive-- I call this a positive angle--
4:41then the average velocity is positive
4:44but if I have a negative angle
4:47then the average velocity would be negative.
4:50For instance, between t4 and t5, if I draw this line
4:56then this angle here is negative
5:00and so the average velocity between t4 and t5
5:03is now negative.
5:08Again, if I had changed the zero points
5:11you would have found the same values
5:13for the average velocity.
5:14The only difference would have been
5:16the position of the curve in that plot.
5:22There is a very big difference in physics
5:24between speed and velocity.
5:28The average velocity between time t1 and t5 is zero
5:33but the average speed is not.
5:37The average speed is defined as the distance traveled
5:46divided by the time that it takes to travel that distance.
5:50Now, what is the distance that the object traveled
5:53between time t1 and time t5?
5:58Well, the object started at a position here on this x axis
6:04and then it went up, reached the highest point here
6:08so I'll make a drawing for you here.
6:10It reached the highest point here, then it went down.
6:15And then when it went here
6:17it went up again and comes down again and it's back.
6:22And in order to find the average speed
6:25you would now have to know exactly what this distance is
6:28add up this distance
6:30add up this distance and this distance.
6:32And if that distance altogether were, for instance, 300 meters
6:37and if the time between t1 and t5 were three seconds
6:43then the average speed
6:44would be 300 meters divided by three seconds.
6:48That would be 100 meters per second
6:50so the average speed would be 100 meters per second
6:52yet the average velocity would be zero.
6:58If you look at the location t3 and t2
7:05and I bring t3 closer and closer to t2
7:11then this angle of alpha will increase
7:14and I can go to the extreme
7:16that I bring t3 almost right at t2.
7:20The angle of alpha will then be tangential to this point.
7:27This will then be my angle of alpha.
7:31And now you will understand how we define
7:35the instantaneous velocity at time t
7:40which is different from an average velocity
7:42between two time intervals.
7:45The instantaneous velocity, v-- and I pick a random time, t--
7:52equals the limiting case
7:55for x measured at time t plus delta t
7:59minus x measured at time t divided by delta t
8:05and I do that for delta t-- goes to zero.
8:09So think of this as being t3 and this as t2.
8:13I bring t3 closer and closer and closer to t2
8:18and the time between them then goes to zero.
8:21And this is something that you undoubtedly recognize.
8:25That's the first derivative of the position versus time.
8:29And now comes an equation
8:31which is one of the very few
8:33that I want you to remember in x... in 801:
8:37v equals dx/dt.
8:41This is one that you must remember, not only in 801
8:44but for the rest of your time at MIT.
8:47And this could be larger than 0, this could be 0
8:51and this could be smaller than 0.
8:53If the angle of alpha, the tangential, is positive
8:56then it is a positive value.
8:58If it is negative, however, when you're here
9:01then it is a negative velocity.
9:03And if the angle of alpha is zero
9:05then it... the velocity is zero.
9:10So if we now look at this plot
9:12we can search for the times that the velocity is zero
9:16so you have to look for the derivative being zero.
9:19That means the angle alpha being zero.
9:22Clearly, here the velocity is zero.
9:26Right here, at this turning point--
9:28that means when the object is here-- it is zero.
9:30When the object is here
9:32it is again zero at this moment in time.
9:34Again, the angle is zero, and it is again zero here.
9:38So those are the times that the velocity is zero.
9:41What are the times that the velocity is positive?
9:44Well, it's positive here.
9:46The velocity's positive here
9:48still positive, positive, becomes negative
9:51negative, positive, zero, negative.
9:55So that's the definition of v, instantaneous velocity.
10:02What is the instantaneous speed?
10:05Well, speed is not sign-sensitive.
10:09Suppose that the velocity here-- just... I call that v1--
10:14suppose that was plus 30 meters per second.
10:18I just grabbed this number out of the blue.
10:21And suppose here, somewhere, it was... I call that v2--
10:26suppose that was minus 100 meters per second.
10:29This is negative and this is positive.
10:31Then we would have to say, in physics--
10:33whether you like it or not, it's not very pleasing--
10:36but you would have to say
10:37that this velocity is lower than that one
10:39because minus 100 is lower than plus 30.
10:43But the speed, of course, is higher here
10:46because the speed is the magnitude of the velocity
10:51and is not sign-sensitive.
10:53So this has the highest speed, of 100 meters per second
10:56and this has a lower speed
10:58but this has the lowest velocity.
11:00It's just an algebraic game
11:02but very important when you make your calculations.
11:07I have always wondered what the average speed
11:12or the average velocity is of a bullet.
11:16Now I want you to realize I am not a fan of guns at all
11:19but it always intrigued me.
11:21How can I measure the average speed of a bullet--
11:26and I have discussed it with some people here--
11:29and we came up with an easy way to do that.
11:33We have a wire, which goes into the blackboard, wire I
11:39and we have another wire that goes
11:40into the blackboards, wire II, and the separation is D meters.
11:48We have to measure that.
11:49The set-up is here
11:50so this is wire number I and this is wire number II.
11:57So you will see D coming in like this,
12:01so I'll make this a I and I'll make this a II.
12:04That's the way it's set up.
12:06And we fire the bullet, which breaks this wire.
12:11At that moment, the timer starts
12:13and then it breaks this wire and that's when the timer stops.
12:19Now, I told you a measurement is meaningless
12:23without knowledge of the uncertainty in your measurement.
12:28So there are two uncertainties involved--
12:31the distance and the timing uncertainty.
12:34This distance I will measure for you, D.
12:40I have here a large ruler.
12:45Here's one wire, here's the other wire.
12:49I cannot do that any better, really
12:51than maybe even half a centimeter
12:54because the situation is not all that stable--
12:56I don't know what happens when the bullet will hit the wire--
12:59so I would say it is 148½ centimeters
13:03but I cannot guarantee it to better than half a centimeter--
13:10148½ plus or minus 0.5 centimeters.
13:16I want you to appreciate
13:18that this is a very small percentage error.
13:22This is only five parts out of 1,500.
13:25That is one out of 300,
13:27so that is only a one-third percent error.
13:30That's very small-- that's what we call the relative error.
13:34Then I ask myself the question--
13:38I want to measure the accuracy of the speed of the bullet
13:41to about two percent.
13:43That was my goal.
13:46How accurate should I do the timing?
13:48Well, I had to make an estimate very roughly
13:53how fast the speed of the bullet is and--
13:56I would think it is probably lower than the speed of sound.
13:59The speed of sound is 340 meters per second.
14:01I don't know whether it's 200 or 300
14:03but it's got to be somewhere in that ballpark
14:05of the kind of bullets that we have--
14:07200 or 300 meters per second.
14:09Let us assume that the speed is 300 meters per second--
14:13just a wild guess.
14:15Then it would take 5 milliseconds
14:18for this bullet to cross from here to here.
14:22And if I want to make a measurement
14:24to two percent accuracy
14:26I have to know this timing
14:28to about one-tenth of a millisecond
14:31because one-tenth of a millisecond
14:34is about two percent of five.
14:36So that sets the accuracy
14:38that I need to make the time measurements.
14:41And so we do have a timer.
14:42It is about accurate to about a tenth of a millisecond
14:45and so now I can measure that time.
14:51So I am going to have here
14:53some time that we measure plus or minus 0.1
14:57and we'll do the whole thing in milliseconds.
15:01But our final answer will be in meters per second.
15:08All right, I always have to think hard when I do this
15:12because when we deal with bullets, that is no kid stuff
15:18and I... as I said
15:20I have really no experience firing guns.
15:24This is the bolt.
15:29There we go.
15:34Here's the bolt.
15:37There we go.
15:43It's in place.
15:44Before I do that, I want to check... check the circuits.
15:47I want to make sure that the electronic circuit
15:49is properly working.
15:52You see the timing here, right?
15:55So I do a small test
15:56just to see whether the circuit is working.
16:00Yep, should be working.
16:04Here comes the bullet.
16:13You ready? I'm ready.
16:15Three, two, one, zero.
16:17(bullet whacks metal)
16:19What do we see?
16:225.8 milliseconds.
16:265.8.
16:28Is that what you see?
16:30Yeah?
16:325.8 milliseconds.
16:365.8 plus or minus 0.1.
16:40So out comes the average.
16:42Call it speed or call it velocity
16:44it's the same thing in this case.
16:47148.5, 5.8, and I have to convert it to meters per second.
16:54That brings it at 256, plus or minus.
17:00Now you come in here, with your plus or minuses.
17:04This is a one point... one-third percent error.
17:06It's negligible to this one.
17:09One out of 58 is about 1.7%
17:12so this is the only one we have to worry about
17:14so the uncertainty in there is about 1.7%.
17:17It's less than two--
17:18that's what I wanted and it gives me an error
17:20of about four meters per second.
17:24And so this is the result.
17:26And you see, it's only meaningful
17:28because we have a good idea about the uncertainties
17:33in the measurement.
17:37Just as we introduced average velocity
17:43now I am going to introduce average acceleration.
17:47Notice that the velocity changes here throughout time.
17:54And that brings me to the next part
17:59the logical part, namely, that we are going to introduce
18:04an average acceleration
18:07and with a little bit of imagination
18:08you can probably guess what that looks like.
18:14The average acceleration between time t1 and time t2
18:20would then be the velocity at time t2
18:22minus the velocity at time t1, divided by t2 minus t1.
18:28And the dimension is lengths per seconds per time squared
18:32so it's meters per second squared.
18:34This is done for a one-dimensional situation.
18:38This number can be larger than zero, it can be equal to zero
18:41and it can be smaller than zero.
18:44In our case, t1 to t2 here
18:49notice the velocity is zero as a start.
18:52And it begins to increase
18:54because this angle of alpha increases.
18:56It's the angle that matters.
18:58The angle increases, so in our case from t1 to t2
19:02the average acceleration is larger than zero.
19:06Look at the angle.
19:08However, if you take the average acceleration between t1 and t5
19:14that is smaller than zero
19:17because here the velocity is zero
19:21but here the velocity is negative.
19:24So if you substitute that in there
19:26you get an average acceleration which is smaller than zero.
19:31So the signs in the velocity
19:32and the signs in average acceleration depend crucially
19:36on how I have defined my increasing value of x
19:39not where I choose my zero points.
19:42If I reverse the direction of increasing x
19:45then all my signs will change.
19:49So you can also write down then
19:51that average acceleration, if you like that
19:53is delta v divided by delta t
19:55but you must be careful because the delta v is sign-sensitive.
19:59You must obey your sign convention.
20:04I have here a tennis ball
20:07and I can bounce this tennis ball, I can throw it down.
20:12And let us assume, just for simplicity
20:14that it hits the floor at about five meters per second
20:18and that it's a very, very good tennis ball
20:21and that it also bounces back
20:24with a speed of about five meters per second.
20:28I will choose this to be my increasing value of x
20:33and so it hits the floor like this.
20:37That means the velocity at which it hits the floor
20:41is minus five meters per second.
20:44It bounces off, there it comes
20:47and it goes up with plus five meters per second.
20:51I call this v1 and I call this v2.
20:56So what, now, is the average acceleration?
20:59Well, I would have to know the time that it takes
21:02for this change in direction.
21:05In other words, we call that the impact time.
21:08I would say, in this case, the impact time delta t
21:12is probably about a hundredth of a second
21:15and so my average acceleration would be v2 minus v1--
21:21that is plus five minus minus five--
21:23that is ten divided by ten to the minus two
21:27and that is plus 1,000 meters per second squared.
21:32I have observed carefully the signs.
21:36If now I say, "Aha, I don't like this
21:39I want to go this-- the value of increasing x."
21:43No big deal.
21:44This will become a plus, this will become a minus
21:47and then this would become a minus.
21:49So then the acceleration
21:50is minus 1,000 meters per second squared.
21:58I have also here a tomato and I have here some eggs.
22:07Now, imagine now that I throw the tomato down
22:11or, for that matter, the egg
22:14and that they hit the floor at five meters per second.
22:18I could do that.
22:19They would not come back up.
22:22They would go... (blows raspberry)
22:24So therefore the change in velocity would not be ten--
22:29apart from the sign that you have to think about--
22:31but it would only be five meters per second.
22:35The impact time would probably be much longer
22:40maybe a quarter of a second.
22:42So therefore the average acceleration during the impact
22:47would then be only five divided by one quarter...
22:51would be something like 20 meters per second squared.
22:56Now, whether you call it plus
22:57or whether you call it minus 20 meters per second squared
23:00depends on your convention of what you call increasing x.
23:04But the eggs and the tomatoes don't care
23:07what you call minus and what you call plus.
23:09Whether the acceleration is
23:11minus 20 meters per second squared
23:13or plus 20 meters per second squared
23:15you'd better believe it, the egg will break.
23:18So it's only in your convention that it matters
23:20but, of course, the physics will not change.
23:24The eggs couldn't care less
23:26what you have chosen for your sign convention.
23:29Something breaks
23:31because the magnitude of acceleration becomes too high.
23:34That's why something breaks.
23:37A few days ago, I saw a Sherlock Holmes movie
23:41and there was a guy who fell on the floor--
23:45marble floor-- hit his head, was lying there motionless.
23:51And here was Watson, and Watson said to Sherlock Holmes
23:55"What happened?"
23:57Sherlock Holmes walks over to the guy
23:59touches him and he says, "He crushed his skull."
24:04He looked very intelligent, I must say, when he said that.
24:07"He crushed his skull."
24:08And I said, "Gee, that's really physics in action--
24:11It's 801 all the way."
24:12(class laughs )
24:13A modest... a really modest velocity when he hits the floor
24:17but he hit the floor like a billiard ball.
24:20The guy was bald, for one thing
24:22and so the impact time was very short.
24:25And when the impact time is short
24:27even if you hit the floor with a modest speed
24:29the acceleration is high... (blows raspberry )
24:31And that was too much
24:33and so that's why his skull was crushed.
24:38So what matters is this changing velocity and the impact time.
24:44We now want to make one last step from average acceleration.
24:50We want to go to the acceleration
24:54at any moment in time
24:56just the way we did that with velocity.
25:00And that now is a natural step.
25:02The acceleration at any moment in time
25:05will be the limit for delta t goes to zero for v
25:12measured at t plus delta t minus vt divided by delta t.
25:21That is the instantaneous acceleration.
25:24And this, you will recognize
25:26is the first derivative of velocity versus time
25:30which is also the second derivative
25:32of position versus time.
25:34And so here comes the second equation
25:36that I really want you to remember
25:39forever and ever and ever
25:41that the acceleration is dv/dt
25:46which is also d2x/dt squared.
25:55We can go to our plot and we can ask ourselves the question now:
26:00where is the acceleration zero, where is it larger than zero
26:04and where is it smaller than zero?
26:05Because this value can be larger than zero, equal to zero
26:09and smaller than zero.
26:12And now you have to be very careful
26:15when you try to derive that from this plot.
26:18You have to be very careful
26:19because you and I have no good feeling for second derivatives.
26:23Velocity is easy--
26:24all you have to do is looking at alpha.
26:27But when it comes to the second derivative
26:28you have to see how alpha is changing.
26:32Well, right here, the velocity is not changing
26:39so the acceleration everywhere here must be zero.
26:43Here the velocity is increasing
26:45so the acceleration must be larger than zero here.
26:50Here, the velocity is almost constant--
26:54it's almost a straight line.
26:56What does that mean for the acceleration?
26:59Zero, exactly.
27:02Here, when it makes this rounding curve
27:04the velocity is positive here, but on this side it's negative
27:07so what does that mean for the acceleration?
27:10Negative, you got it.
27:12And so you can now roughly find
27:14where the acceleration is positive
27:17where it's negative and where it is zero.
27:24Let's do a straightforward example
27:27the way that you could expect it on an assignment
27:31or, if you were extraordinarily lucky
27:35you might even get something like that on an exam.
27:39Very straightforward.
27:41I'm going to give you the position x as a function of time
27:46and then ask you lots of questions about it.
27:50So this example is a working example--
27:53x equals eight minus 60 plus t-squared.
28:04So this tells you where the object is at any moment in time
28:08and let this be in meters.
28:12What now is the velocity at any moment in time?
28:15Well, that's the derivative dx/dt
28:19and I use the following-- x equals t to the power n.
28:27Then, as most of you should know
28:30dx/dt is then n times t to the power n minus one.
28:35That's all I'm using.
28:37So the derivative of eight is zero.
28:39I get here minus six, I get here plus 2t--
28:46this would be in meters per second---
28:48and the acceleration...
28:50I have to take the derivative of the velocity, I get plus two.
28:55So notice that the acceleration
28:57is constant in time, is not changing
29:00but the velocity is changing.
29:04Well, at time t equals zero...
29:09just, I will start to probe a little bit.
29:11I want to get a feeling for what this object is doing.
29:14At time t equals zero, x is plus eight
29:18The velocity is minus six meters per second
29:23and the acceleration equals plus two.
29:29I can also ask myself at what time does x = 0?
29:34What are the times that x is zero?
29:36Well, I have to solve this second-order equation
29:39which is something that you've all done in high school
29:42and you will find that that's the case
29:44when the time is plus two and when the time is plus four.
29:51Take the plus two... that makes this four.
29:564 + 8 = 12, minus 6 x 2, that's 0.
30:01So you see the 2 works and you check that the 4 also works.
30:06Just for my curiosity, when is the velocity zero?
30:09Oh, that's easy--
30:10that's when this equation is zero
30:12so that's when t equals three.
30:16What is, at that moment, the position?
30:18Oh, I substitute t equals three in here
30:21and that gives me minus one.
30:24x = -1.
30:28So now I'm ready to plot x as a function of t.
30:33It's, of course, a parabola
30:35and I use this information that we have just derived.
30:39So here comes my plot.
30:46Let this be increasing value of x
30:50let this be eight and let this be minus one.
30:56This is the time axis.
31:00I have a zero here
31:02and so I want to cover, let's say, about six seconds
31:07so I have 1, 2, 3, 4, 5, 6.
31:18Now I am going to use this information
31:21in order to give you a curve which is similar to that one
31:25except this is a simple one-- this is just a parabola.
31:29So I know that at time t equals zero
31:32the object is at position eight.
31:35I know that x is zero... that x is zero
31:40at the time 2 and at the time 4
31:43so the object is here at this time and at this time.
31:48And I know that at time t equals three
31:51it is at position minus one.
31:58And I also know that the velocity is zero
32:01so we can check that.
32:03And so if I make this plot now
32:06then we would get a curve that's sort of like this
32:11and yes, indeed, notice, the velocity here is zero.
32:15The angle alpha equals zero.
32:21The object starts out at t equals zero
32:24with a negative velocity.
32:27You can see that-- the object at t equals zero is here.
32:32This is where the object is, I hope you realize that.
32:34The object is never here.
32:35This is the road, this is the one-dimensional track
32:38on which the object is sitting.
32:40The object is here and it starts going in this direction.
32:43If it starts going in this direction
32:45the velocity must be less than zero
32:47and indeed it is, it's minus six.
32:50But there is the acceleration
32:53which is plus two in this direction.
32:55The acceleration says, "I don't want you to go down.
32:59I want you to go up!"
33:01Well, the velocity says
33:03"Sorry, all I can do is slowly, slowly change"
33:06and that's what it's doing.
33:08It's slowly changing the velocity
33:10and there comes a time that the velocity is zero
33:13so the object goes down, the velocity changes
33:17and when it is at position minus one
33:19it has come to a grinding halt and now it is returning.
33:23This positive value of a is now increasing the velocity
33:27and that's what you see.
33:29I therefore bet you a nickel
33:31that if you substitute, in that equation, t equals four
33:35that the velocity better be positive.
33:38It has changed from a minus sign to a plus sign
33:41because of this positive acceleration.
33:43I bet you a nickel t equals four.
33:48What is x... uh, what is v?
33:50We want to know v.
33:528 - 6 + 2 meters per second.
33:55You see?
33:56Physics works-- v is now plus two meters per second.
34:00So all that information is in there
34:02but I want you to be able to also digest it.
34:05Don't look at that curve
34:07as just some dumb parabola, some dumb curve.
34:10Try to imagine what is happening
34:12and only then do you get some insight.
34:15Then you really begin to get it in your brains.
34:20I now would like to write down, in most general form
34:25the equation for the position and the velocity
34:30as a function of time for a one-dimensional motion
34:35whereby the acceleration is constant.
34:38So it's going to be one-dimensional again
34:41and we have a is going to be a constant.
34:44And so the equation that I write down
34:46is the most general way that I can write it down.
34:50So we're going to get x equals some number C1
34:55plus some C2 times t, plus some C3 times t squared.
35:02And notice... oh, I already erased my example.
35:05My example is gone
35:06but you would have seen this was an eight before
35:08and here we had... uh, what did we have?
35:11Minus... we had minus 6t and we had plus 1t squared.
35:18So you recognize these three... I can now take the derivative
35:23and so I get C2 plus 2C3 times t
35:32and then I get the acceleration equals 2C3.
35:38And now we get some insight into these quantities.
35:44Clearly, x1... C1 is the position of x
35:50at time t equals zero
35:52for which we often write an x zero.
35:55Because when t is zero, that is where x is.
35:59C2 is really the velocity at time t equals zero
36:05because when t is zero, that's when C2 is v.
36:10And the acceleration is now changing with time.
36:15It's 2C3, therefore C3 is half the acceleration.
36:22So this gives you some insight
36:23in the meaning of these quantities
36:25and you can see... you can read now, some physics in there.
36:29C1, C2, and C3 can independently be
36:32either zero, or larger than zero, or negative.
36:35It makes no difference-- each one of these combinations
36:38is a valid possibility in physics.
36:44When we have gravity
36:46an object is influenced by the gravitational acceleration
36:53and the gravitational acceleration is a constant.
36:57And we write, often
36:59for that gravitational acceleration, the letter "g".
37:03Whether I drop an object or throw it vertically up
37:08or I throw it vertically down, it's all one-dimensional.
37:11It becomes two-dimensional when I throw it at an angle.
37:14I keep it one-dimensional
37:16the acceleration is always the same
37:19and that g-- gravitational acceleration--
37:22in Boston is 9.80 meters per second squared
37:28and it varies a little bit for different places on Earth.
37:32This gravitational acceleration is independent
37:37of the mass of the object that I drop
37:40of the speed of the object
37:42of the chemical composition of the object
37:45of the size of the object and of the shape of the object
37:48assuming that we have no air drag
37:50assuming that these experiments are done in... in vacuum.
37:55Is it obvious that the gravitational acceleration
37:59is independent of all these quantities?
38:01By no means.
38:04Is it true?
38:05We think so, but I want you to appreciate
38:08that it is not obvious
38:09and it can not be proven from first principles.
38:16Remember, last time we dropped an apple from three meters
38:20and we dropped another one from one and a half meters.
38:25And in your second assignment, which you haven't seen yet
38:28I'm asking you to calculate
38:30the gravitational acceleration for me
38:32using these both experiments.
38:35And, of course, I want you to also tell me
38:37what the uncertainty is in your final answer.
38:40And I'd like to help you a little bit to set it up
38:46and also to get these equations in terms of gravity.
38:52Whenever we deal with gravity, we get the g in there.
38:57So suppose here is the object at time t equals zero.
39:02It was the apple, and I call that position x zero.
39:05I call that zero, I'm free to choose my zero position
39:08and I drop it zero speed.
39:10I just let it go, because that's the way we did it in class.
39:14The object goes down and it hits the floor.
39:20Well, the general equations, now, which deal in gravity...
39:26If I call this the increasing value of x...
39:30You can choose it differently.
39:31This is my choice today... is the following.
39:35x equals x zero plus v zero t plus one-half g t squared
39:46and g now is 9.80 meters per second squared.
39:54The velocity, at any moment in time
39:57equals v zero plus gt
40:01and the acceleration is constant-- it's simply g.
40:06Now, in my case, I have chosen t equals zero, x zero, zero
40:10and I have chosen this zero, so these go.
40:15And so you see that when the object is here--
40:18which is three meters below this point--
40:21and you know the time, how long it took to get there
40:24that you can now calculate "g"
40:26because x would be then three meters.
40:28That's when it's here.
40:30We made a measurement in class
40:31how long it took, so you know the time
40:34and so you can come up with a value for g.
40:36And you can do that for both measurements
40:39and, of course, I want you to tell me, also
40:42what the uncertainty is in those measurements.
40:46Remember that we derived, last time, that C...
40:51that the time that it takes for the apple to fall
40:54was C times the square root of h over g
40:56and we never knew what that C was.
40:59I did a demonstration to show you
41:01that the time is proportional to the square root of h.
41:04We never knew what that C was.
41:06Now you know, because now you have the equations here
41:09and you see that that C simply was the square root of two.
41:12But I could not derive that from my dimensional analysis.
41:20Now I want you to relax and, at the same time
41:26get a little bit alert for a change.
41:30Look at this situation, v equals gt.
41:34That means when I drop an apple--
41:36and I'm going to drop another one today--
41:38that the velocity increases with time.
41:42So if I strobe this apple while it was falling
41:46I would see the separation, when it strobes
41:50to increase with time, because the velocity goes up with time.
41:57I have here an apple, or I am going to put an apple up
42:00about three meters from the floor-- three meters.
42:06So the height is three meters, approximately.
42:11We know from last time, remember, we did it
42:13it was about 780 milliseconds to hit the floor.
42:16I will just round it off and I think about it...
42:20about eight-tenths of a second, just to get an idea.
42:24If I flash it, if I strobe it twice per second--
42:30we call that two hertz--
42:32so my strobe is two times per second.
42:39Then I should hit that ball, when it's falling
42:42twice with my strobe light.
42:45I don't know where it is, though
42:47because when we strobe it and when I let the apple go
42:50the two are not synchronized, so maybe the first time
42:53that the light blinks, it may be here
42:56and the second time, it may be here.
42:58But it's also possible that the first time it's here
43:00and the second time, it's there.
43:02And so the first thing I want to do
43:04is to test your alertness.
43:06We will blink.
43:08You will tell me where you see them.
43:11But we will take a picture.
43:14We will take a picture which will show us
43:16exactly where those two balls were.
43:18So that's the first alertness test.
43:20So get ready for this, and then we will do a second one
43:24which is even more intriguing.
43:27So now I have to first lower this velvet
43:36so that we get a nice dark background.
43:46There we go.
43:48(whooshes )
43:55Wow, with my fingerprints on it, it's not so black any more.
44:01There it is... that's the background.
44:11Oh, what am I doing?
44:12I need the ladder again-- I have to bring the apple up!
44:17Friday's always a bad day for me.
44:20Okay... so now I am going to bring the apple up.
44:24There's some metal here, there are electromagnets
44:29and so I throw a switch here
44:31so that the electromagnet is activated.
44:34Very similar to what we did last time.
44:37We have to put the apple up and the apple is hanging there.
44:42There we go.
44:51So now I have to start the, uh...
44:59The strobe.
45:06That's about two hertz, that's about two flashes per second
45:09and I'm going to make it pitch black.
45:15Pitch black.
45:23All the lights go off.
45:26I will count down 3, 2, 1, 0
45:30and Bob, there, who is behind the camera
45:32will open the shutter when I say "one."
45:35And when I say "zero", the ball will fall.
45:39So you may only see the ball in its highest position.
45:44That may not count there, of course
45:45because it makes two flashes in the time
45:48that the shutter is open and that I drop it.
45:51Okay, if you're ready, I'm ready.
45:54Make it as dark as we can.
45:57Bob, are you ready?
46:01Class ready?
46:02CLASS: Yes.
46:03LEWIN: Everyone ready?
46:04You don't look ready.
46:06LEWIN: Okay... three, two, one.
46:12That was zero.
46:14So let's look at this again in slow motion.
46:22So now we are developing the picture
46:25and I would like you to tell me where you saw the balls.
46:32Where were they, roughly?
46:35Where was the first one?
46:36How much... how much below the highest point?
46:40Only this much?
46:43The first one.
46:44And then the second one was pretty low, then.
46:47(class murmurs )
46:48Okay, sounds interesting.
46:50We'll take a look.
46:52While the picture is developing
46:56I'm now going to test your real alertness.
47:01I'm going to strobe it with an unknown frequency...
47:06unknown to you.
47:10I will tell you a secret-- it's a higher frequency.
47:13You're going to see more balls on the way down.
47:16I'm not going to ask you where they are, exactly.
47:20All I want you to tell me, afterwards, how many you saw.
47:24That's all.
47:26So count them as it falls.
47:28You know we have only 0.8 seconds to count.
47:33Bob, how did the picture come out?
47:39Wow, you're good!
47:41Whoa, you're good.
47:43It was very high, actually...
47:45the first... the first flash, very high.
47:56You see, it's... you did very well.
47:59We're going to start, now, with the second part.
48:13Is the audio restored?
48:15Should be.
48:17So, I activated the magnet again.
48:26There it is.
48:36Oh, goodness!
48:43Working?
48:45Okay, thank you, Bob.
48:47Okay, Bob, if you're ready, I'm ready.
48:51We're going to make it as dark as we can.
48:54So all I want you to tell me, how many balls will you see?
48:58Alright, ready?
49:00Bob, you're ok?
49:02Three, two, one...
49:07(class laughing )
49:09Well?
49:14Who saw three?
49:17Four?
49:21Four, I want to know four.
49:23Five?
49:25Five, here's a five, there's a five.
49:29Another five?
49:31Who saw six?
49:33Wow... seven?
49:36Eight?
49:37Nine?
49:39Ten?
49:40Eleven?
49:42Who just saw a blur?
49:45(class laughs )
49:46Those are the real winners, I think.
49:49Well, I'll tell you, it was ten hertz.
49:51Since it was 0.8 seconds, depending upon where you hit it
49:55how lucky you are, I will show you.
49:57You will either see seven or maybe eight balls
50:02but it was a good test.
50:05And for those of you who thought that it was only...
50:11that only saw five, there you see them, let's count them.
50:16Let's count them together.
50:22One, this is one.
50:24Two, three, four, five, six, seven, this is a bounce.
50:29So for those who saw five, I would say
50:31"Take some rest this weekend, you need it"
50:33and I'll need it, too.
50:34See you Monday.

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