This lecture is going to be about the effect of air resistance on projectiles.
Air resistance is the force of air on objects moving through it. The more
velocity an object has, the more air pushes back against the object's
velocity. So in this case we can imagine the baseball getting faster and faster
through the air in this direction and as a result the force of air resistance
gets bigger and bigger on the ball. So air resistance is always working
perfectly against the direction of the velocity. So if the velocity is pointing
in this direction, the air resistance will push back up in this direction. This
creates a pattern where the faster an object is moving through the air, the
more it is accelerated in the opposite direction because air resistance is
pushing it back and there's usually nothing pushing it forwards. So if it
starts off moving fast it's rapidly slowed down because there's a lot of
force of air resistance on it, but as it loses velocity the air resistance also
goes down, which means that the force slowing the object down is itself
getting smaller, so it actually slows down at a lower and lower rate, so it
takes longer to slow down if you're moving at a slower velocity through the
air, so the rate of slowing itself slows down as you slow. A lot to take in there!
Hopefully this visual helps a little bit. We can see the effect of air resistance
on the shape of the path of projectiles. This is an example of an object moving
without air resistance, and the noticeable thing about an object moving
without air resistance is that it forms the shape of a perfect parabola. This is
because its motion in the X Direction is constant and its motion in the Y
Direction is symmetrical. Whatever it's doing on one side, it's doing the
opposite on the other side, and so another thing that's noticeable here is
that the displacement for the first half of the object's motion in the X
direction is perfectly equal to the X Direction displacement in the second
half of its motion. It's symmetrical in that way, so if you see an object that's
moving parabolically that's perfectly symmetrical, you know that there's no air
resistance applying on the object. Now let's apply air resistance to this ball
and see how its motion changes so you'll notice that the path is different. It
doesn't go up as high and it doesn't travel as far and you'll also notice
that it travels farther in the first half of its motion than it does in the
second half of its motion. This is because as it moves forward the velocity
vector is being slowed down by the force of air resistance so that means that
it's moving slower in its second half of motion compared to
the first so in the second half of its motion it moves less far in the
x-direction than the first and the force of air resistance is also stopping it
from going up as high. So you can see that the displacement in the X direction
in the first half of its motion is much larger than the displacement in the
second half of its motion, and it doesn't achieve the same height, so that will be
the shape of projectiles with air resistance. We can also graph the motion
of objects that are experiencing air resistance with position, velocity, and
acceleration graphs. We normally only pay attention to the y-direction of an
object that's experiencing air resistance when we're graphing it. The
x-direction graphs don't really come up as much, so I'm only going to focus on
those for this lecture. Let's start with an object that's free falling through
the air or it starts in freefall but as it moves through the air, air resistance
begins to pick up so in the second that it's dropped it accelerating down at
9.81 meters per second, because in the second it's dropped it doesn't have any
velocity yet, so the only force acting on it is gravity because no velocity means
no air resistance up, and if the only force acting on it is gravity that means
that it starts with 9.81 m/s^2 of acceleration, and i'm considering the
downward motion to be positive here which is why that's on the positive part
of my graph, but as it falls it starts to pick up velocity very quickly because it
has a large downward acceleration, so as it picks up velocity quickly it also
picks up air resistance very quickly, but as it gets more and more air resistance
it's also getting less and less total net force because the air resistance is
working against the force of gravity, so the less net force it has the less
acceleration it has in the downward direction, which means that it takes
longer to get more velocity, which means it takes longer to get more air
resistance, so the more air resistance you have the harder it is to get even
more, so that's why the acceleration graph here is curving down. It starts off
very easy to get more air resistance, so the acceleration rapidly drops off, but
as you keep going it becomes harder and harder to get more air resistance and
add to that. This continues in this pattern until you hit a point where the
acceleration finally becomes zero where the air resistance balances out the
force of gravity, and when this happens this is terminal velocity, this is where
the ball is moving at its fastest possible velocity and can't get any
faster, because the acceleration is zero because the forces are balanced out, so
that point at the end of the graph is where terminal velocity occurs for the
acceleration graph. We can also look at the velocity in the y-direction for this
object. We can imagine again that it's dropped so that means that it starts at
zero and as it accelerates, it starts off gaining a lot of velocity in a little
time, but as air resistance begins to apply to it it becomes harder and harder
to get additional velocity until the air resistance perfectly balances out the
force of gravity. One way that we can understand this is to refer back to our
acceleration graph, so I put the acceleration graph below the velocity
graph to the left of my head so you can look at what the graph is doing as you
look at what the velocity graph is doing. Remember that the slopes of the tangent
lines on a velocity graph represent the acceleration, so if the acceleration
starts very positive but rapidly decreases, that means that the slopes of
the tangent lines are also going to start very positive but rapidly decrease,
but as they keep going they're decreasing at a slower and slower rate,
so they're getting to zero at a slower and slower rate as the graph goes on,
until at the very end it finally hits zero. So the point where the acceleration
is zero that's where the velocity graph will have a flat slope, so it's just
staying at that constant velocity, so that point there that constant velocity
is the terminal velocity of the object. That y-axis value is the terminal
velocity of the object. We can also look at the acceleration and velocity graphs
of objects that are moving from one medium to another medium. Let's say for
example that we dropped this ball above the ocean and see what happens as it
falls, and then when it hits the water it moves into the water, so the second that
we drop it it starts off with an acceleration of 9.81 because the only
force on it is gravity, and it starts with a velocity of zero, but just like in
our previous graphs these two graphs curve in this way until it hits its
terminal velocity. So at the line that I drew that's where terminal velocity
happens on the graph, and that Y value on the velocity graph is the exact terminal
velocity of the ball in the air, so it hasn't hit the water yet, so it's going
to continue to fall at that same terminal velocity, so the velocity is
staying constant which means that the acceleration is staying at zero and then
at this moment is where the ball hits the water, and when the ball hits the
water it's no longer feeling the force of air resistance pushing it up. It's now
experiencing a buoyant force which is the name for the force from water, and
water pushes back on things much more strongly than air does. You know it's
much easier to move your hand through air that it is through water so that
means that suddenly there's a much larger force pushing up on the ball than
there is pushing down so that means that the net force is pointing up, so that
means that the acceleration is also pointing up on this ball, which means
that on my graph the acceleration is suddenly going to jump into the negative
value because we're considering down to be positive, so up is negative for the
rest of the ball's trip. It's going to slow down because the acceleration is
pointing up while the ball moves down through the water, so that means that the
acceleration is taking away from the velocity, so the buoyant force is going
to shrink, because as you slow down as the acceleration takes away from you
you're moving through the water less quickly, so the water is putting a
smaller and smaller force back on you, so this is what the next part of the graph
would look like, so it continues to slow until the buoyant force balances out the
force of gravity and when it does that the ball is now at a new smaller
terminal velocity because it was slowing that entire time, and the forces are
balanced, so the acceleration is now zero, so this is the new terminal velocity of
the ball in the water. And if nothing interrupts that it will just continue to
fall at that rate until it hits the bottom, so obviously that graph is very
complicated. If you watch this a few more times and try to take in every little
bit of information there's no one part of it that's especially challenging. It's
just understanding everything that's happening that's a lot to take in at
once. This exact same graph can also apply to other physical situations.
Skydivers are really popular in physics when talking about air resistance so
when the skydiver dives out of the plane the only force on them in the moment
they dive out is the force of gravity because they're not moving yet so
there's no air resistance so that means that their acceleration starts at 9.81
and as they begin to fall the force of air resistance gets bigger and bigger
and bigger so those first parts of the graph are the same as before until the
skydiver hits a terminal velocity without their parachute, and then they
can follow that terminal velocity for a certain amount of time, and let's say
this is the point where their parachute opens, and the parachute makes their
force of air resistance much larger. It has such a large surface area, so
suddenly the net force is pointing up, which means that the acceleration is
also pointing up, which means that because on my graph up is negative the
acceleration is suddenly going to jump to being very negative. So now that the
velocity is still positive but the acceleration is negative, the
acceleration is going to rapidly take away from the velocity, and as the
velocity gets smaller the force of air resistance gets smaller until it
balances out with the force of gravity again, and by that time the velocity has
gotten significantly smaller, so now the person is at a new terminal velocity
with their parachute that they will maintain until they hit the ground. This
is the reason why people use parachutes when they skydive. It's because when you
open the parachute you achieve a much smaller terminal velocity so you can
land safely. You're not gonna have to do specific math with these types of graphs.
The important thing is to understand why the patterns in them are the way that
they are. Why they're curving in certain spots. Why they're staying flat in
certain parts, and what those curves and flat parts mean in relation to the
forces and the velocities