section 12.3 – velocity and acceleration
DESCRIPTION
Section 12.3 – Velocity and Acceleration. Vector Function. A vector function is a function that takes one or more variables and returns a vector: Where and are called the component functions. A vector function is essentially a different notation for a parametric function. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Section 12.3 – Velocity and Acceleration](https://reader035.vdocument.in/reader035/viewer/2022062520/5681614b550346895dd0cbf1/html5/thumbnails/1.jpg)
Section 12.3 – Velocity and Acceleration
![Page 2: Section 12.3 – Velocity and Acceleration](https://reader035.vdocument.in/reader035/viewer/2022062520/5681614b550346895dd0cbf1/html5/thumbnails/2.jpg)
Vector Function
A vector function is a function that takes one or more variables and returns a vector:
Where and are called the component functions.
A vector function is essentially a different notation for a parametric function.
![Page 3: Section 12.3 – Velocity and Acceleration](https://reader035.vdocument.in/reader035/viewer/2022062520/5681614b550346895dd0cbf1/html5/thumbnails/3.jpg)
Particle MotionIn AP Calculus AB, particle motion was defined in functions of time versus motion on a horizontal or vertical line.
In AP Calculus BC, particle motion will ALSO be defined in functions of position versus position (along a curve).
How successful you are with Particle Motion is a good predictor of how successful you will be on the AP Test.
![Page 4: Section 12.3 – Velocity and Acceleration](https://reader035.vdocument.in/reader035/viewer/2022062520/5681614b550346895dd0cbf1/html5/thumbnails/4.jpg)
Position Vector FunctionWhen a particle moves on the xy-plane, the coordinates of its position can be given as parametric functions:
for
The particle’s position can also be expressed as a position vector function:
The coordinates of the parametric function at time
t…
A vector function is essentially a different notation for a parametric function.
… are equal to the components of the vector
function at time t.
![Page 5: Section 12.3 – Velocity and Acceleration](https://reader035.vdocument.in/reader035/viewer/2022062520/5681614b550346895dd0cbf1/html5/thumbnails/5.jpg)
Example 1Let the position vector of a particle moving along a curve is defined by (a) Find and graph the position vector of the particle at
.
-4 -3 -2 -1 1 2 3 4
-3
-2
-1
1
2
3 y
x
4 4 43cos ,2sins
2.121,1.414𝑠( 𝜋4 )
![Page 6: Section 12.3 – Velocity and Acceleration](https://reader035.vdocument.in/reader035/viewer/2022062520/5681614b550346895dd0cbf1/html5/thumbnails/6.jpg)
Velocity Vector FunctionThe vector function for position is differentiable at if and have derivatives at .
The derivative of , , is defined as the velocity vector:
A vector function is essentially a different notation for a parametric function.
![Page 7: Section 12.3 – Velocity and Acceleration](https://reader035.vdocument.in/reader035/viewer/2022062520/5681614b550346895dd0cbf1/html5/thumbnails/7.jpg)
Example 1 ContinuedLet the position vector of a particle moving along a curve is defined by (b) Find the velocity vector.
-4 -3 -2 -1 1 2 3 4
-3
-2
-1
1
2
3 y
x
ddtv t s t
3sin ,2cost t
𝑠( 𝜋4 ) 3cos , 2sind ddt dtt t
![Page 8: Section 12.3 – Velocity and Acceleration](https://reader035.vdocument.in/reader035/viewer/2022062520/5681614b550346895dd0cbf1/html5/thumbnails/8.jpg)
Example 1 ContinuedLet the position vector of a particle moving along a curve is defined by (c) Find and graph the velocity vector of the particle at
.
-4 -3 -2 -1 1 2 3 4
-3
-2
-1
1
2
3 y
x
4 4 43sin ,2cosv
2.121,1,414 𝑠( 𝜋4 )
𝑣 ( 𝜋4 )
If the initial point of the velocity vector is also the
terminal point of the position vector, the velocity vector is
tangent to the curve.
![Page 9: Section 12.3 – Velocity and Acceleration](https://reader035.vdocument.in/reader035/viewer/2022062520/5681614b550346895dd0cbf1/html5/thumbnails/9.jpg)
Acceleration Vector Function
The second derivative of , , is defined as the acceleration vector:
A vector function is essentially a different notation for a parametric function.
![Page 10: Section 12.3 – Velocity and Acceleration](https://reader035.vdocument.in/reader035/viewer/2022062520/5681614b550346895dd0cbf1/html5/thumbnails/10.jpg)
Example 1 ContinuedLet the position vector of a particle moving along a curve is defined by (b) Find the acceleration vector.
-4 -3 -2 -1 1 2 3 4
-3
-2
-1
1
2
3 y
x
ddta t v t
3cos , 2sint t
𝑠( 𝜋4 ) 3sin , 2cosd ddt dtt t
![Page 11: Section 12.3 – Velocity and Acceleration](https://reader035.vdocument.in/reader035/viewer/2022062520/5681614b550346895dd0cbf1/html5/thumbnails/11.jpg)
Arc Length and SpeedConsider a particle moving along a parametric curve. The distance traveled by the particle over the time interval is given by the arc length integral:
On the other hand, speed is defined as the rate of change of distance traveled with respect to time, so by the Second Fundamental Theorem of Calculus:
dsdtSpeed
0
2 2'( ) '( )t
ddt t
x u y u du 2 2'( ) '( )x t y t
This is the magnitude of the velocity vector!
![Page 12: Section 12.3 – Velocity and Acceleration](https://reader035.vdocument.in/reader035/viewer/2022062520/5681614b550346895dd0cbf1/html5/thumbnails/12.jpg)
Speed with a Vector Function
The particle’s speed is the magnitude of , denoted :
Speed is a scalar, not a vector.
![Page 13: Section 12.3 – Velocity and Acceleration](https://reader035.vdocument.in/reader035/viewer/2022062520/5681614b550346895dd0cbf1/html5/thumbnails/13.jpg)
ReminderFrom AP Calculus AB:
Speed is the absolute value of velocity:
Integrating speed gives total distance traveled:
Example: If , find the speed at and the total distance traveled during .
THE
SA
ME
AS
VE
CTO
R
FUN
CTIO
NS
!
![Page 14: Section 12.3 – Velocity and Acceleration](https://reader035.vdocument.in/reader035/viewer/2022062520/5681614b550346895dd0cbf1/html5/thumbnails/14.jpg)
White Board ChallengeA particle moves along a curve so that and . What is the speed of the particle when .
222( ) 6 ln dyddt dtv t t t
2 2112 sin 2tt t
2 212(2) 12 2 sin 2 2v
24.672
![Page 15: Section 12.3 – Velocity and Acceleration](https://reader035.vdocument.in/reader035/viewer/2022062520/5681614b550346895dd0cbf1/html5/thumbnails/15.jpg)
Example 2A particle moves along the graph of , with its x-coordinate changing at the rate of for . If , find
(a) The particle’s position at .
You can use the FTOC on components:
2
12 1 dx
dtx x dt 3
22
11
tdt 2 2
1
11
t
2 21 12 1
1 1.75
![Page 16: Section 12.3 – Velocity and Acceleration](https://reader035.vdocument.in/reader035/viewer/2022062520/5681614b550346895dd0cbf1/html5/thumbnails/16.jpg)
Example 2 ContinuedA particle moves along the graph of , with its x-coordinate changing at the rate of for . If , find
(b) The speed of the particle at .
22 dydxdt dtv t
Find the speed equation: Since y is a function of x, we need to use the Chain Rule:
2dy dx dxdt dt dtx
3 32 22t t
x 3 3 3
2 22 2 22t t t
x Substitute this and dx/dt into the speed equation.
3 3 3
2 22 2 22 2 2
2 ___ Substitute t=2
2v 1.75
From (a), we know x=1.75 when t=2
1.152
We know dx/dt.
![Page 17: Section 12.3 – Velocity and Acceleration](https://reader035.vdocument.in/reader035/viewer/2022062520/5681614b550346895dd0cbf1/html5/thumbnails/17.jpg)
Example 3A particle moves a long a curve with its position vector given by for . Find the time when the particle is at rest.
0 0,0v t
The particle is at rest when the velocity vector is:
Find the velocity vector: 2 43cos , 5sind t d tdt dtv t
3 52 2 4 4sin , cost t
Solve:3 52 2 4 4sin 0 cos 0t t
0,2 ,4 2t t is when the particle is at rest because it is the only time on the interval when BOTH components are 0.
![Page 18: Section 12.3 – Velocity and Acceleration](https://reader035.vdocument.in/reader035/viewer/2022062520/5681614b550346895dd0cbf1/html5/thumbnails/18.jpg)
SummaryIf is the position vector of a particle moving along a smooth curve in the xy-plane, then, at any time t,
1. The particle’s velocity vector is ; if drawn from the position point, it is tangent to the curve.
2. The particle’s speed along the curve is the length of the velocity vector, .
3. The particle’s acceleration vector is , is the derivative of the velocity vector and the second derivative of the position vector.