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Vector Calculus Parametric equations – 5.6 Vector Calculus Parametric equations – 5.6

Vector Calculus Parametric equations – 5.6 - PowerPoint Presentation

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Vector Calculus Parametric equations – 5.6 - PPT Presentation

Jacplus Connections to the Study Design AOS 4 Vectors Vector Calculus Position vector as a function of time and sketching the corresponding path given including circles ellipses and hyperbolas in Cartesian and parametric forms ID: 702637

particle vector worked time vector particle time worked find position function equation path cartesian curve speed velocity parametric acceleration

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Slide1

Vector CalculusSlide2

Parametric equations – 5.6 Jacplus

Connections to the Study Design:

AOS

4 – VectorsVector CalculusPosition vector as a function of time , and sketching the corresponding path given , including circles, ellipses and hyperbolas in Cartesian and parametric forms

 Slide3

Parametric Equations

Parameter:

Path traced out by the particle is defined in terms of a third variable,

.In a two dimensional case the path is described by two parametric equations, as both the and coordinates depend upon the parameter, .

 

Recap:

The position vector of the particle is given by

, where

and

are unit vectors in the and directions.Also called a vector function of the scalar real variable , where is called the parameter, and often represents time.

 

If we can eliminate the parameter from the two parametric equations and obtain an equation of the form , this is called an explicit relationship or the equation of the path.It may not be possible to obtain an explicit relationship, but often an implicit relationship of the form can be formed.Careful consideration needs to be given to the possible values of , which then specify the domain (the values) and the range (the values) of the equation of the path.

 Slide4

Parametric Equations - Background

A locus is a set of points traced out in the plane, satisfying some geometrical relationship.

The path described by a moving particle forms a locus and can be described by a Cartesian equation.

Note: The Cartesian equation does not tell us where the particle is at any particular time.The path traced out by the particle can be defined in terms of a third variable – in this case we will use the variable as the parameter.

 

A position vector is given by

, where

and

are unit vectors in the

and directions, this is also called the vector equation of the path.If we can eliminate the parameter from these two parametric equations and obtain an equation of the form

, then this is called an explicit relationship and is the equation of the path.At times it may be unable to obtain an explicit relationship but an implicit relationship can be obtained in the form of

.Either way, the relationship between and is called the Cartesian equation of the path. Slide5

Worked Example 19

Given the vector equation

, for

, find and sketch the Cartesian equation of the path, and state the dome and range.

 Slide6

Eliminating the parameter

Methods to use:

Direct substitution

Trigonometric formulasWorked Example 20Given the vector equation

for

, find and sketch the Cartesian equation of the path, and state the domain and range.

 Slide7

Parametric Representations and Sketching parametric curves

The parametric representation of a curve is not necessarily unique.

To sketch parametric curves, CAS calculators can be used to draw the Cartesian equation of the path from the two parametric equations, even if the parameter cannot be eliminated.

Worked Example 22Show that the parametric equations

and

where

represent the hyperbola

.

 Slide8

Position vectors as functions of time – 13.2

Jacplus

Position vector as a function of time

, and sketching the corresponding path given , including circles, ellipses and hyperbolas in Cartesian and parametric forms

 Slide9

Worked example 1 - Cambridge

Find the Cartesian equation for the graph represented by each vector function:

 

For a vector function

:

The

domain

of the Cartesian relation is given by the range if the function

.

The range of the Cartesian relation is given by the range of the function .In the Example 1b, the domain of the corresponding Cartesian relation is the range of the function

, which is

. The range of the Cartesian relation is the range of the function

, which is

Note:

the Cartesian equation

can be written as

; it is the circle with centre

and radius 1.

 Slide10

Worked Examples - Cambridge

Worked Example 2

Find the Cartesian equation of each of the following. State the domain and range and sketch the graph of each of the relations.

 

Worked Example 3

For each of the following, state the Cartesian equation, the domain and range of the corresponding Cartesian relation and sketch the graph:

 Slide11

Closest approach

Given the position vector of a particle,

, where

and are unit vectors in the and

directions, it is possible to find the positon or coordinates of the particle at a given value of t.It is also possible to find the value of

and the coordinates when the particle is closest to the origin.

 

Worked Example 1 -

Jacplus

A particle moves so that its vector equation is given by for .Find the distance of the particle from the origin when .

Show that the distance of the particle from the origin at any time is

Hence find the closest distance of the particle from the origin. Slide12

Collision problems

Worked Example 2 -

JacPlus

Two particles move so that their position vectors are given by

and

for

. Find:

When and where the particles collide.

Where their paths cross.The distance between the particles when

. Slide13

Position vectors as a function of time – 12B Cambridge

Connections to the Study Design:

AOS 4 – Vectors

Vector CalculusPosition vector as a function of time , and sketching the corresponding path given , including circles, ellipses and hyperbolas in Cartesian and parametric forms

 Slide14

Consider the following scenarios:

A particle travelling at a constant speed along a circular path with radius length

and centre

.The path is represented in Cartesian form as:

If the particle starts at the point

and travels anticlockwise, taking

units of time to complete one circle, then its path is represented in

parametric form

as:

This is expressed in vector form as:

where

is the position vector of the particle at time

.

 

The graph of a vector function is the set of points determined by the function

as

varies.

In two dimensions, the

and

axis

are used.

In three dimensions, three mutually perpendicular axes are used. It is best to consider the

and

axes as in the horizontal plane and the

axis as vertical and through the point of intersection of the

and

axes

.

 Slide15

Information from the vector function

The vector function gives much more information about the motion of the particle than the Cartesian equation of its path.

For example, the vector function

indicates that:

At time

, the particle has position vector

. That is, the particle starts at

The particle moves with constant speed on the curve with equation

.

The particle moves in an anticlockwise directionThe particle moves around the circle with a period of

, ie. It takes

units of time to complete one circle.The vector function describes a particle moving anticlockwise around the circle with equation

, but this time the period is

.

The vector function

again describes a particle moving around the unit circle, but the particle starts at

and moves clockwise

.

 

Worked Example 4 - Cambridge

Sketch the path of particle where the position at time t is given by

, where

 Slide16

Notes:

The equation

, gives the same Cartesian path, but the rate at which the particle moves along the path is different.

 

Motion in two dimensions:

When a particle moves along a curve in a plane, its position is specified by a vector function of the form

Motion in three dimensions:

When a particle moves along a curve in three-dimensional space, its position is specified by a vector function of the form

 

If

, the again the Cartesian equation is

, but

Hence the motion is along the curve shown and in the direction indicated.

 Slide17

Worked Examples - Cambridge

Worked Example 5

An object moves along a path where the position vector is given by

Describe the motion of the object.

 

Worked Example 6

The motion of two particles is given by the vector functions

and

where

.

Find:The point at which the particles collideThe points at which the two paths crossThe distance between the particles when  Slide18

Vector Calculus – 13.3 and 13.5

jacplus

Connections to the Study Design:

AOS 4 – VectorsVector CalculusDifferentiation and anti-differentiation of a vector function with respect to time and applying vector calculus to motion in a plan including projectile and circular motionSlide19

Vector Calculus

Consider the curve defined by

Let

and be points on the curve with positions vectors

and

respectively.

Then

It follows that

is a vector parallel to

.

As

, the point Q approaches P along the curve. The derivative of r w.r.t is denoted by and is defined by:

provided that this limit exits.

The vector

points along the tangent to the curve at

, in the direction of increasing

.

Note:

The derivative of a vector function

is also denoted by

or

.

Additionally:

A unit tangent vector at

is denoted by

.

 Slide20

Rules for differentiating vectors

Derivative of a constant vector

If

is a constant vector, that is a vector which does not change and is independent of t, then .Note that

 

Derivative of a sum or difference of vectors

The sum or difference of two vectors can be differentiated as the sum or difference of the individual derivatives.

That is,

and

Using these rules, if then

Simply put, to differentiate a vector we merely differentiate each component using the rules for differentiation.

 Slide21

Derivative of a vector functions

Two Dimensions

, only if

and

are differentiable.

 

First Derivative:

Second Derivatives:

 

Three Dimensions

 

First Derivative:

Second Derivatives:

 Slide22

Properties of the derivative of a vector function

, where

is a constant vector

, where

is a real number

, where

is a real-valued function

 Slide23

Worked Examples - Cambridge

Worked Example 7

Find

and if

.

Worked Example 8

Find

and

if

.

Worked Example 9

If find and

, where

.

Worked Example 10

If

find

and

, where

 

Worked Example 11

A curve is described by the vector equation

.

Find:

Find the gradient of the curve at the point

, where

and

Worked Example 12

A curve is described by the vector equation

, with

.

Find the gradient of the curve at the point (

x,y

), where

and

.

Find the gradient of the curve where

.

 Slide24

Worked Example 3 - Jacplus

Find a unit tangent vector to

at the point where

.

 Slide25

Derivative summary

Velocity vector

Because

represents the position vector,

represents the velocity vector.

Note

the single dot above r indicates the derivative w.r.t time.

Furthermore, if

, then

.

 

SpeedThe speed of a moving particle is the magnitude of the velocity vector. The speed at time is given by

.

If the particle has a mass of

, then the magnitude of the momentum acting on the particle is given by

.

 

Since

represents the velocity vector, differentiating again w.r.t

gives the acceleration vector.

The acceleration vector is given by

.

Note

that the two dots above the variables indicate the second derivative w.r.t time.

 

Acceleration vectorSlide26

Worked example 4 - jacplus

A particle spirals outwards so that its position vector is given by

for

.

Find the velocity vector.

Show that the speed of the particle at time

is

and hence find the speed when

.

Find the acceleration vector.  Slide27

Worked example 7 - Jacplus

The position vector,

, of a golf ball at a time

seconds is given by

for

, where the distance is in metres,

is a unit vector horizontally forward and

is a unit vector vertically upwards above ground level.

Find when the golf ball hits the ground

Find where the golf ball this the groundDetermine the initial speed and angle of projectionFind the maximum height reachedShow that the golf ball travel in a parabolic path. Slide28

Antidifferentiation

When integrating a function, always remember to include the constant of integration, which is a scalar. When integrating a vector function w.r.t a scalar, the constant of integration is a vector.

This follows since if

is a constant vector, then

.

 

The constant VectorSlide29

Integration of vector functions

Integrating a Velocity

Vector w.r.t time

gives a Position Vector

Note:

that an initial condition must be given in order for us to be able to find the constant vector of integration.

 

Integrating an Acceleration Vector to gives a Position Vector

Given the acceleration vector

,

w

e can obtain the velocity vector

w

here

is a constant vector.

 Slide30

Worked examples - cambridge

Example 13

Given that

, find:

if

if also

 

Example 14

Given

with

and

, find . Slide31

Worked examples - jacplus

Worked Example 10

The velocity vector of a particle is given by

for

. If

, find the position vector at time

.

 

Worked Example 11

The acceleration vector of a particle is given by , where is the time. Given that and

, find the position vector at time

. Slide32

Worked examples - jacplus

Worked Example 12

The acceleration vector of a moving particle is given by

for

, where

is the time. The initial velocity is

and the initial position is

.

Find the Cartesian equation of the path.

 Slide33

Velocity and Acceleration for motion along a curve – 12D Cambridge

Connections to the Study Design:

AOS 4 – Vectors

Vector CalculusPosition vector as a function of time , and sketching the corresponding path given , including circles, ellipses and hyperbolas in Cartesian and parametric forms

 Slide34

Velocity

Consider a particle moving along a curve in the plane, with position vector at time

given by:

We can find the particle’s velocity at time as follows. Acceleration

Acceleration is __________________Therefore a(t), the acceleration __________________, is given by

Recap – fill in the blanks

Velocity is ______________________

Therefore

, the velocity ____________, is given by:

The velocity gives _______________________

 

Speed

Distance between two points on the curve

Speed is __________________________________.

At time t, the speed is denoted by _________________

The _______________ distance between two points on the curve is found using:Slide35

Worked Examples

Worked Example 15

The position of an object is

metres at time seconds, where

.

Find at time

:

The velocity

The acceleration vector

The speed Worked Example 16The position vector of a particle at time

is given by

, where .Find:The velocity of the particle at time

The speed of the particle at time

The minimum speed of the particle

 

Worked Example 18

The position vector of a particle at time

is given by

The velocity at time

The speed of the particle at time

The maximum speed

The minimum speed

 Slide36

Worked Example - Cambridge

Worked Example 17

The position of a projectile at time t is give by

for

where

i

is a unit vector in a horizontal direction and j is a unit vector vertically up.

The projectile is fired from a point on the ground.

Find:The time taken to reach the ground againThe speed at which the projectile hits the ground

The maximum height of the projectileThe initial speed of the projectile Slide37

Worked Examples - Cambridge

Worked Example 19

The position vectors, at time

, of particles and are given by

Prove that

and

collide while travelling at the same speed but at right angles to each other.

 Slide38

Worked examples - Cambridge

Worked Example 20

A particle moves along a line such that its position at time

is given by the vector function:

How far along the line does the particle travels from

to

?

 

A particle moves along a curve such that its position vector at time

is given by:

How far along the curve does the particle travel from

to

(Give your answers to three decimal places.)

Find the shortest distance between these two points.

 

Worked Example 21