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## Two-dimensional projectile motion

# Projectile on an incline

## Video transcript

What I want to do
in this video is tackle a problem that would
be considered pretty difficult for most first-year
physics students. And you, frankly,
probably wouldn't be expected to solve
a problem like this in most first-year
physics class. Or if you're in an
advanced or honors class you might be expected or it
might be a bonus problem. But it's an interesting
type of problem. Because what we're
going to do is, we're going to launch a
projectile on an incline. So maybe we're on
the side of the hill. So it's a hill. Let me do it in green. So let's say we're on the
side of the hill like this. And let's say that we know
the inclination of the hill. The hill's inclination is 30
degrees off the horizontal. So this is the horizontal
right over here. So that is the
inclination of the hill. And then we're going
to launch a projectile at 10 meters per second. We're going to launch it
at 10 meters per second. And the angle with the
hill is 15 degrees. So at a 15 degree
angle with the hill. And the reason why
this is more difficult than the traditional projectile
motion problems is, well, we could think about it. The projectile is
going to be launched and it is going to eventually
land at some point on the hill. But we can't do
the simple figure out how long it's in the air
using its vertical velocity because we don't know what
the vertical displacement for this thing is
going to be, unless we know how far down
the hill it lands. Because the further
down the hill it lands, the higher the
vertical displacement. So we have to think
about both the horizontal and the vertical displacement
at the same time. And as we walk
through this, you'll see how that can be done. So I guess the first thing
that we really would always want to do whenever
you want to try to solve this type of problem is
break up our velocity into both the horizontal and
vertical components. So the vertical
component of our velocity is going to be the magnitude
of our total velocity, 10 meters per second, times--
and be very careful here-- not the sine of 15 degrees,
but the sine of the angle with the horizontal. So times the sine of 45 degrees. And I go into it in a lot more
detail in previous videos. For the sake of time,
I won't go into it. But it really just
comes from sohcahtoa. If we were to draw the
vertical component, it would look like this. This is the angle. The sine of 45 degrees
is equal to the opposite over the hypotenuse or
the hypotenuse times the sine of 45 degrees is equal
to the vertical component. That's where it's coming from. Let me get rid of some
of the stuff I just drew just so it makes
it a little bit cleaner. And the horizontal component
of our velocity is going to be, by the same logic, 10
cosine of 45 degrees. Now let's think about what
the horizontal displacement is going to be. And I'm just going to go
straight to the formula that we've derived in
the last few videos. The horizontal
displacement is going to be our initial-- sorry. Let's do the vertical
displacement. The vertical
displacement-- I could have done the horizontal
displacement first-- but the vertical
displacement is going to be our initial
vertical velocity, which we know as 10 sine
of 45 degrees. And by the way, we could
just solve that right now. What is the sine of 45 degrees? Sine of 45 degrees is
square root of 2/2. Cosine of 45 degrees is
also square root of 2/2. So both of these values,
10 times square root of 2/2 is 5 square roots of 2. So this whole thing
right over here is 5 square roots of
2 meters per second. That's our vertical velocity. And our horizontal
velocity is also 5 square roots of 2
meters per second. So that simplifies
things a little bit. But anyway, we
were talking about our vertical displacement. Our vertical
displacement is going to be our initial vertical
velocity, 5 square roots of 2, times our change in time
plus the acceleration. Well, we know what
the acceleration is. It's negative 9.8 meters
per second squared. So let me write minus 9.8. I'm not going to write the
units here so that we say space. Times our change
in time squared. All of that over 2. We derive this in several
videos, especially the last few where we do these two
dimensional projectile motions. So this gives us
our displacement in the y direction. And I can simplify
this a little bit. Our displacement in
the vertical direction is equal to 5 square
roots of 2 times delta t times-- let me do it in
the same-- times delta t change in time minus 4.9 times
our change in time squared. So we know we have this
constraint right over here. So this gives us our
vertical displacement as a function of time. Let's think about our
horizontal displacement as a function of time. Our horizontal
displacement is going to be equal to our
horizontal velocity, which is 5 square roots of 2
times our change in time. Now what can we do next? Well, we have to have
some relationship between our horizontal
displacement and our vertical displacement. And that relationship
is going to be given to us by this incline. So wherever we
land-- let's say this is where we eventually
do end up landing. Well, let's think
about our horizontal and our vertical
displacements and what their relationship
would have to be. So if this is where
we land, then this would be our-- let me do it
in the same colors-- that right there would be my
vertical displacement. I would move that far up. And then our
horizontal displacement will be this right over
here, will be that length right over there. So that is our
horizontal displacement. So what is the
relationship between our vertical displacement and
our horizontal displacement? And we know that this angle
right over here is 30 degrees. So we can use some
basic trigonometry. We have a right triangle. We know the opposite
side from the angle. We know the adjacent side. And the trig function that uses
the opposite and the adjacent is the tangent function. So we get the
tangent of 30 degrees is going to be equal
to the magnitude of our vertical displacement
over the magnitude of our horizontal displacement. And the tangent of
30 degrees, that's the same thing as
the sine of 30 over-- let me just do it over here. So the tangent of 30
degrees is the same thing as the sine of 30 degrees
over the cosine of 30 degrees. Let me do this a
little bit neater. And the sine of
30 degrees is 1/2. This is equal to 1/2. And the cosine of 30 degrees
is the square root of 3/2. So this is equal to 1/2 times
2 over the square root of 3, which is equal to 1 over
the square root of 3. So we get the magnitude
of our vertical component over the magnitude of our
horizontal component-- is our horizontal
component right over here-- is equal to 1 over
the square root of 3. What's useful about
this is it gives us a relationship
between our horizontal and vertical component,
or between our vertical and our horizontal components. And we can use this
constraint right over here to then solve for
one of these two. And let me show you
how we'll do it. So let's just
explicitly write this. Well, let's do it this way. Let us cross multiply
here, which is really the same thing as multiplying
both sides by the square root of 3 and the magnitude of
our horizontal component. We get the square root
of 3 times the magnitude. And both of these are
going to be positive. Well, let me just
write it this way. Times the magnitude of
our vertical component is going to be equal
to the magnitude of our horizontal component. So right, just like that. So we now have a relationship
between the length of these two vectors. And we can use this
relationship to substitute back into the constraints
that we already have. So the second constraint
right over here, let's use this information. The second constraint says
that our horizontal component of our displacement is
equal to 5 square roots of 2 times our change in time. Or another way of
thinking about it, if we divide both sides
by 5 square roots of 2, we get our change
in time is equal to the horizontal component
of our displacement divided by 5 square roots of 2. But we also know that
the horizontal component of our displacement
is the square root of 3 times the vertical
component of our displacement. Here I explicitly wrote
the magnitude notation. When we start dealing with
either just the vertical or the horizontal
component, I can just write it like this,
because it's either going to be a positive
or a negative value. And that specifies both the
magnitude and the direction. So what I'm going to do right
over here, and obviously the way I've drawn it right
over here, both of these are going to be positive values. It's upwards displacement
in the vertical direction, so that's positive
by our convention. And we're moving to the right. So that's positive,
also, by our convention. So I can rewrite this
over here as being equal to square root of 3 times
our vertical displacement. And all of that's over
5 square roots of 2. Now, the whole
reason why I did this is, this expression right here
contains this information, contains the ratio between
our vertical displacement and our horizontal displacement. And it also contains
the information of, how does the horizontal
displacement, how does that change as a function of time? So our time needs
to be equal to this. So this is our
time as a function of our vertical
displacement now, not time as a function of
our horizontal displacement. And what we can do is, we
can use this constraint with our original
vertical displacement as a function of
time to then solve for our vertical displacements. So let's do that. Let's substitute this
business right here for delta t in this top
equation right over here. So if we do that-- and
I'll write it big-- we get our vertical
displacement, right over there, is equal to 5 square
roots of 2 times delta t. So it's 5 square roots of 2. Delta t is all of this
business over here. So 5 square roots
of 2 times delta t. Delta t is square root of 3
times our vertical component. All of that, really
the magnitude of our vertical component,
over 5 square roots of 2. So that's that right there. Or actually, we could look
at this one right over here. We could use this constraint. This is just simplified. So then we have minus 4.9
times delta t squared. So delta t squared is
this quantity squared. I'll just write it out. I don't want to
skip too many steps. So delta t once again is
the square root of 3 times the vertical component. All of that over 5 square
roots of 2 squared. And now what does this give us? So now we literally have
a quadratic equation with only one variable. So we can solve for this. But let me rewrite it. Let me simplify it. So we have our
vertical component is equal to-- now we have
a 5 square roots of 2 in the numerator and
one in the denominator and they cancel out. So we get the square root of 3
times our vertical component, the magnitude of our
vertical component. It's actually also specifying,
well, the magnitude, we can say, for now, although
I'm not using that notation. So then we have minus 4.9
times this quantity squared. So that's going to be, the
square root of 3 squared is 3 times the vertical
component squared. And then over 5 squared,
which is 25, times 2. That's the square
root of 2 squares. So 25 times 2 is 50. And so we get, if we were to
simplify this a little bit more. Let's subtract this
from both sides. We get-- I'll do it
all in one color-- 0 is equal to the square
root of 3 minus 1 times our vertical component. Because if we subtract
this from both sides, that's square root of 3
times our vertical component minus 1 times our
vertical component. So that's the square
root of 3 minus 1 times our vertical component. And then we have all
of this business. Minus 4.9 times 3 over 50 times
our vertical component squared. And lucky for us, we
can just factor out one of these s sub y's over
here, one of these vectors. And so we can get-- and I'll
just do that in place so that I don't-- well, let me just-- I
don't want to skip many steps. So this is equal to the square
root of 3 minus 1, minus 4.9. Let me do it in that color. So let me do it like this. It's equal to the square root
of 3 minus 1 minus 4.9 times 3 over 50 times one of these,
times our vertical component. And then we factored one of
those vertical components out. So we factored one of them out. So the vertical component
of our displacement could either be-- so we have
the product of two things that equal 0. So our vertical
displacement could be 0, which is true, because
at some point in the path we literally had 0
vertical displacement. That was literally
where we started. But that's not the
answer we're looking for. We're looking for this
vertical displacement. So either this is going to be 0. But that's just kind
of the obvious answer. Or all of this business
is going to be equal to 0. But this is pretty easy
to solve for 0 over here. So we get square root
of 3 minus 1 minus 4.9. Let me just calculate
all of these things just so that I don't have
to keep writing them. So we get the square root of
3 minus 1 is equal to 0.73205. So I'll just write 0.732 here. So this is equal to 0.732. That's that part
right over there. And then 4.94. I know this problem
is getting long. Just pause it and take a
break if you're getting tired. 4.9 times 3 divided by
50 is equal to 0.294. So minus 0.294. I could put a 0 out front just
to make it clear where we are. Times this, times our
vertical component. This could also be equal to 0. Either this is 0 or that is 0. When this is 0, it gives
us the obvious answer. We're more interested in this. To solve for this, we can
add this to both sides. And we get 0.732 is
equal to negative 0.294 times the vertical component. We are in the home stretch. We divide both sides by
this, by negative 0.29. Oh sorry, this will
now be positive. I almost made a careless mistake
after 16 minutes of video. So now we divide
both sides by 0.294 and we get our
vertical displacement. So this, I think, a drum
roll might be in order. So we have 0.732 and
all of this business, but I'll just round it there,
divided by 0.294 gives us a vertical displacement of,
if we round, 2.50 meters. Or 2.49 meters, I should say. So this is equal to 2.49 meters. This is exciting. This is equal to 2.49 meters. And now we can figure out
the horizontal displacement pretty easily,
because we know that the horizontal displacement
is square root of 3 times the vertical displacement. So let's figure that out. That's the vertical
displacement. Let's multiply that times
the square root of 3. And we get 4.31 meters. So we get the
horizontal displacement. The horizontal displacement
is equal to 4.31 meters. So this is equal to 4.31 meters. So we actually now know
the total displacement in the vertical direction and
in the horizontal direction. And I'll leave it up to you. If you wanted to figure out
exactly how far along the hill we traveled, you
can just use both of these values in the
Pythagorean theorem to essentially figure
out the hypotenuse of this right triangle.