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## Average and instantaneous acceleration

Current time:0:00Total duration:9:06

# Acceleration

## Video transcript

In this video, I want to talk a
little bit about acceleration. And this is probably
an idea that you're somewhat familiar
with, or at least you've heard the term
used here or there. Acceleration is just the
change in velocity over time. Probably one of the most typical
examples of acceleration, if you're at all
interested in cars, is that many times
they will give you acceleration numbers, especially
for sport cars, actually all cars if you look
up in Consumer Reports, or wherever they give the
stats on different cars. They'll tell you something like,
I don't know, like a Porsche-- and I'm going to make up
these numbers right over here. So let's say that we
have a Porsche 911. They'll say that a Porsche 911,
they'll literally measure it with a stopwatch, can go
0 to 60 miles per hour. And these aren't
the exact numbers, although I think it's
probably pretty close. 0 to 60 miles per hour
in, let's say, 3 seconds. So, although officially what
they're giving you right here are speeds, because
they're only giving you magnitude and no
direction, you can assume that it's in
the same direction. I mean, We could
say, 0 miles per hour to the east to 60 miles per
hour to the east in 3 seconds. So what was the
acceleration here? So I just told you the
definition of acceleration. It's change in
velocity over time. So the acceleration-- and
once again acceleration is a vector quantity. You want to know not only
how much is velocity changing over time, you also care
about the direction. It also makes sense
because velocity itself is a vector quantity. It needs magnitude
and direction. So the acceleration
here-- and we're just going to assume
that we're going to the right, 0 miles per
hour and 60 miles per hour to the right-- so it's going
to be change in velocity. So let me just write it
down with different notation just so you could
familiarize yourself if you see it in the
textbook this way. So change in velocity. This delta symbol right
here just means "change in." Change in velocity over time. It's really, as I've
mentioned in previous videos, it's really time is
really a change in time. But we could just
write time here. This 3 seconds is
really change in time. It might have been, if you
looked at your second hand, it might have been 5 seconds
when it started, and then my 8 seconds when it stopped, so
it took a total of 3 seconds. So time is really a
change in seconds. But we'll just go with time
right here, or just with a t. So what's our
change in velocity? So our final velocity
is 60 miles per hour. And our original velocity
was 0 miles per hour. So it's 60 minus
0 miles per hour. And then, what is our time? What is our time over here? Well, our time is, or we could
even say our change in time, our change in time is 3 seconds. So this gives us 20 miles
per hour, per second. Let me write this down. So this becomes,
this top part is 60. 60 divided by 3 is 20. So we get 20. But then the units are
little bit strange. We have miles. Instead of writing MPH, I'm
going to write miles per hour. That's the same thing as MPH. And then we also, in the
denominator, right over here, have seconds. Which is a little bit strange. And as you'll see, the
units for acceleration do seem a little bit strange. But if we think it
through, it actually might make a little
bit of sense. So miles per hour. And then we could either
put seconds like this, or we could write per second. And let's just think
about what this is saying, and then we could get
it all into seconds, or we could all get into
hours, whatever we like. This is saying that every
second, this Porsche 911 can increase its velocity
by 20 miles per hour. So its acceleration is 20
miles per hour, per second. And actually, we should
include the direction, because we're talking
about vector quantities. So this is to the east. So this is east, and then
this is east right over here-- just so we make sure that
we're dealing with vectors. You're giving it a
direction, due east. So every second it can
increase in velocity by 20 miles per hour. So hopefully, with
the way I'm saying it, it makes a little bit of sense. 20 miles per hour, per second. That's exactly what
this is talking about. Now we could also
write it like this. This is the same thing
as 20 miles per hour, because if you take
something and you divide by seconds, that's the
same thing as multiplying it by 1 over seconds. So that's miles
per hour-seconds. And although this
is correct, to me this makes a little
less intuitive sense. This one literally says it. Every second, it's increasing in
velocity by 20 miles per hour. 20 miles per hour increase
in velocity per second. So that kind of
makes sense to me. Here it's saying 20
miles per hour-seconds. So once again, it's
not as intuitive. But we can make this so it's
all in one unit of time, although you don't
really have to. You can change this
so that you get rid of maybe the hours
in the denominator. And the best way to get rid
of an hour in the denominator, is by multiplying
it by something that has hours in the numerator. So hour and seconds. And here, the smaller
unit is the seconds. So it's 3,600 seconds
for every 1 hour. Or 1 hour is equal
to 3,600 seconds. Or 1/3600 of an hour per second. All of those are legitimate
ways to interpret this thing in magenta
right over here. And then you multiply, do a
little dimensional analysis. Hour cancels with hour. And then will be
equal to 20/3600. 20/3600 miles per
seconds times seconds. Or we could say,
miles-- let me write it this way-- miles per
seconds times seconds. Or we could say,
miles per second-- I want to do that in
another color-- miles per second squared. And we can simplify
this a little bit. Divide the numerator
and denominator by 10. You get 2/360. Or you could get,
this is the same thing as, 1/180 miles
per second squared. And I'll just
abbreviate it like that. And once again, this 1/180
of a mile, how much is that? You might want to
convert to feet. But the whole point
in here is, I just wanted to show you
that, well, one, how do you calculate acceleration? And give you a little bit
of a sense of what it means. And once again, this
right here, when you have seconds squared in
the bottom of your units, it doesn't make a ton of sense. But we can rewrite
it like this up here. This is 1/180 miles per second. And then we divide by
seconds again, per second. Or maybe I can write
like this, per second, where this whole thing
is the numerator. So this makes a
little bit more sense from an acceleration
point of view. 1/180 miles per
second, per second. Every second, this
Porsche 911 is going to go 1/180 of a
mile per second faster. And actually, it's
probably more intuitive to stick to the miles per
hour, because that's something that we have a little
bit more sense on. And another way to visualize it. If you were to be
driving that Porsche, and you were to look at the
speedometer for that Porsche, and if the acceleration
was constant-- it's actually not going to
be completely constant-- and if you look at
speedometer-- let me draw it. So this would be 10,
20, 30, 40, 50, 60. This is probably not what
the speedometer for a Porsche looks like. This is probably more analogous
to a small four cylinder car's speedometer. I suspect the Porsche's
speedometer goes much beyond 60 miles per hour. But what you would
see for something accelerating this fast is,
right when you're starting, the speedometer
would be right there. And that every second it would
be 20 miles per hour faster. So after a second
the speedometer would have moved this far. After another second
the speedometer would have moved this far. And then after another
second the speedometer would have moved that far. And the entire time
you would have kind of been pasted to the
back of your seat.

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