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## Solving motion problems using parametric and vector-valued functions

Current time:0:00Total duration:6:22

# Motion along a curve: finding velocity magnitude

AP Calc: FUN‑8 (EU), FUN‑8.B (LO), FUN‑8.B.1 (EK)

## Video transcript

- [Voiceover] A particle moves
along the curve xy equals 16, so that the y coordinate is increasing, we underline this, the y
coordinate is increasing at a constant rate of
two units per minute. That means that the rate of
change of y with respect to t is equal to two. What is the magnitude,
in units per minute, of the particle's velocity vector when the particle is at the point
four comma four, so when x is four, and y is four. So let's see what's going on. So, let's first just
remind ourselves what a velocity vector, what the
velocity vector will look like. So our velocity is going
to be a function of time. And it's going to have two components, it's going to be what is
the rate of change in the x direction, and the rate of
change in the y direction. So the rate of change in
the x direction is going to be dx dt, and the rate of
change in the y direction is going to be dy dt. And they tell us that this
is, that dy dt is a constant two units per minute. But they're not even
just asking us for just the velocity vector for its components, they're asking for the magnitude, they're asking for the
magnitude of the particle's velocity vector. Well, if I have some vector,
let me do a little bit of a side here, if I have
some vector, let's say a, that has components b and
c, well then the magnitude of my vector, sometimes you'll
see it written like that, sometimes you'll see it
written with double bars like that, the magnitude of my vector, and this comes straight out
of the Pythagorean theorem, this is going to be the
square root of b squared plus c squared. The square root of the
x component squared plus the y component squared. So, if we wanted the magnitude
of our velocity vector, the magnitude of the
particle's velocity vector, well I could write that
as the magnitude of v, I could even write it as a function of t, it's going to be equal to the square root of the x component squared,
so that's the rate of change of x with respect to time, squared, plus the y component
squared, which in this case is the rate of change of y,
with respect to t, squared. So how do we figure out
these, how do we figure out these two things? Well we already know
the rate of change of y with respect to t, they say
that's a constant rate of two units per minute. So we already know that
this is gonna be two, or that this whole thing right
over here's gonna be four, but how do we figure
out the rate of change of x with respect to t? Well, we could take our
original equation that describes the curve, we could take
the derivative of both sides with respect to t, and
then that's going to give us an equation that involves
xy and dx dt and dy dt, so let's do that. So we have xy is equal to 16, I'm gonna to take the
derivative with respect to t, of both sides, we do that
in a different color, just for a little bit of variety, so the derivative with respect
to t of the left hand side, derivative with respect to
t of the right hand side, now the left hand side,
we view this as a product of two functions, if we say,
look, x is a function of t, and y is also a function
of t, this is we're gonna do a little bit of the product rule and a little bit of the chain rule here. And so this is going to be equal to, derivative of the first
function, which is, so, we'll first say the derivative
of x with respect to x is one, times the derivative
of x with respect to t. Remember, taking the derivative
with respect to t, not with respect to x, times
the second function, so times, times the second function, so times y, times y,
plus, the first function, which is just x, times the
derivative of the second function with respect to t. So first what's the derivative
of y with respect to y, well that's just one, and
then what's the derivative of y with respect to t, well that's dy dt, and that is going to be equal to, that is going to be equal
to, derivative of a constant is just zero. So let's see, what does this simplify to, this simplifies to, in
fact we don't even have to simplify it more,
we can actually plug in the values to solve for dx dt. We know that dy dt is a constant two, and we want the magnitude
of the particle's velocity vector when the
particle's at the point four comma four, so
when x is equal to four, so when x is equal to four,
and y is equal to four, and y is equal to four. So now, it's a little messy right now, but this right here is an
equation we can solve for, there's only one unknown here, the rate of change of x with respect to t, right when we are at the
point four comma four, and so if we're able to figure that out, we can substitute that
in here and figure out the magnitude of our velocity vector. So let us write it out, so this gives us four, four dx dt, plus what is this four times two, plus eight is equal to
zero, and so we have four dx dt is equal to negative eight, just subtracted eight from both sides, divide both sides by four, you get dx dt, scroll down, is equal to negative two. So when all this stuff
is going on the rate of change of x with respect
to t is negative two, and then you square it, you
get a four, right over here, and so the magnitude
of our velocity vector is going to be equal to the
square root of four plus four, which is equal to eight,
which is the same thing as four times two, so this is going to be two square root of two, units per minute, so that's the magnitude
of the velocity vector.

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