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
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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