VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn...

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VECTOR FUNCTIONS VECTOR FUNCTIONS 13

Transcript of VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn...

Page 1: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

VECTOR FUNCTIONSVECTOR FUNCTIONS

13

Page 2: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

13.4Motion in Space:

Velocity and Acceleration

In this section, we will learn about:

The motion of an object

using tangent and normal vectors.

VECTOR FUNCTIONS

Page 3: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Here, we show how the ideas of tangent

and normal vectors and curvature can be

used in physics to study:

The motion of an object, including its velocity and acceleration, along a space curve.

MOTION IN SPACE: VELOCITY AND ACCELERATION

Page 4: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

In particular, we follow in the footsteps of

Newton by using these methods to derive

Kepler’s First Law of planetary motion.

VELOCITY AND ACCELERATION

Page 5: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Suppose a particle moves through

space so that its position vector at

time t is r(t).

VELOCITY

Page 6: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Notice from the figure that, for small values

of h, the vector

approximates

the direction of the

particle moving along

the curve r(t).

VELOCITY

( ) ( )t h t

h

r r

Vector 1

Page 7: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Its magnitude measures the size

of the displacement vector per unit

time.

VELOCITY

Page 8: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

The vector 1 gives the average

velocity over a time interval of

length h.

VELOCITY

Page 9: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Its limit is the velocity vector v(t)

at time t :

VELOCITY VECTOR

0

( ) ( )( ) lim

'( )

h

t h tt

ht

r rv

r

Equation 2

Page 10: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Thus, the velocity vector is also

the tangent vector and points in

the direction of the tangent line.

VELOCITY VECTOR

Page 11: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

The speed of the particle at time t

is the magnitude of the velocity vector,

that is, |v(t)|.

SPEED

Page 12: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

This is appropriate because, from Equation 2

and from Equation 7 in Section 13.3,

we have:

= rate of change

of distance with

respect to time

SPEED

| ( ) | | '( ) |ds

t tdt

v r

Page 13: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

As in the case of one-dimensional motion,

the acceleration of the particle is defined as

the derivative of the velocity:

a(t) = v’(t) = r”(t)

ACCELERATION

Page 14: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

The position vector of an object moving

in a plane is given by:

r(t) = t3 i + t2 j

Find its velocity, speed, and acceleration when t = 1 and illustrate geometrically.

VELOCITY & ACCELERATION Example 1

Page 15: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

The velocity and acceleration at time t

are:

v(t) = r’(t) = 3t2 i + 2t j

a(t) = r”(t) = 6t I + 2 j

VELOCITY & ACCELERATION Example 1

Page 16: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

The speed at t is:

VELOCITY & ACCELERATION Example 1

2 2 2

4 2

| ( ) | (3 ) (2 )

9 4

t t t

t t

v

Page 17: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

When t = 1, we have:

v(1) = 3 i + 2 j

a(1) = 6 i + 2 j

|v(1)| =

VELOCITY & ACCELERATION Example 1

13

Page 18: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

These velocity and acceleration vectors

are shown here.

VELOCITY & ACCELERATION Example 1

Page 19: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Find the velocity, acceleration, and

speed of a particle with position vector

r(t) = ‹t2, et, tet›

VELOCITY & ACCELERATION Example 2

Page 20: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

VELOCITY & ACCELERATION Example 2

2 2 2 2

( ) '( ) 2 , , (1 )

( ) '( ) 2, , (2 )

| ( ) | 4 (1 )

t t

t t

t t

t t t e t e

t t e t e

t t e t e

v r

a v

v

Page 21: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

The figure shows the path of the particle in

Example 2 with the velocity and acceleration

vectors when t = 1.

VELOCITY & ACCELERATION

Page 22: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

The vector integrals that were introduced in

Section 13.2 can be used to find position

vectors when velocity or acceleration vectors

are known, as in the next example.

VELOCITY & ACCELERATION

Page 23: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

A moving particle starts at an initial position

r(0) = ‹1, 0, 0›

with initial velocity

v(0) = i – j + k

Its acceleration is

a(t) = 4t i + 6t j + k

Find its velocity and position at time t.

VELOCITY & ACCELERATION Example 3

Page 24: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Since a(t) = v’(t), we have:

v(t) = ∫ a(t) dt

= ∫ (4t i + 6t j + k) dt

=2t2 i + 3t2 j + t k + C

VELOCITY & ACCELERATION Example 3

Page 25: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

To determine the value of the constant

vector C, we use the fact that

v(0) = i – j + k

The preceding equation gives v(0) = C. So,

C = i – j + k

VELOCITY & ACCELERATION Example 3

Page 26: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

It follows:

v(t) = 2t2 i + 3t2 j + t k + i – j + k

= (2t2 + 1) i + (3t2 – 1) j + (t + 1) k

VELOCITY & ACCELERATION Example 3

Page 27: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Since v(t) = r’(t), we have:

r(t) = ∫ v(t) dt

= ∫ [(2t2 + 1) i + (3t2 – 1) j + (t + 1) k] dt

= (⅔t3 + t) i + (t3 – t) j + (½t2 + t) k + D

VELOCITY & ACCELERATION Example 3

Page 28: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Putting t = 0, we find that D = r(0) = i.

So, the position at time t is given by:

r(t) = (⅔t3 + t + 1) i + (t3 – t) j + (½t2 + t) k

VELOCITY & ACCELERATION Example 3

Page 29: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

The expression for r(t) that we obtained

in Example 3 was used to plot the path

of the particle here for 0 ≤ t ≤ 3.

VELOCITY & ACCELERATION

Page 30: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

In general, vector integrals allow us

to recover:

Velocity, when acceleration is known

Position, when velocity is known

VELOCITY & ACCELERATION

0 0( ) ( ) ( )

t

tt t u du v v a

00( ) ( ) ( )

t

tt t u du r r v

Page 31: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

If the force that acts on a particle is known,

then the acceleration can be found from

Newton’s Second Law of Motion.

VELOCITY & ACCELERATION

Page 32: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

The vector version of this law states that if,

at any time t, a force F(t) acts on an object

of mass m producing an acceleration a(t),

then

F(t) = ma(t)

VELOCITY & ACCELERATION

Page 33: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

An object with mass m that moves in

a circular path with constant angular speed ω

has position vector

r(t) = a cos ωt i + a sin ωt j

Find the force acting on the object and show that it is directed toward the origin.

VELOCITY & ACCELERATION Example 4

Page 34: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

To find the force, we first need to know

the acceleration:

v(t) = r’(t) = –aω sin ωt i + aω cos ωt j

a(t) = v’(t) = –aω2 cos ωt i – aω2 sin ωt j

VELOCITY & ACCELERATION Example 4

Page 35: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Therefore, Newton’s Second Law gives

the force as:

F(t) = ma(t)

= –mω2 (a cos ωt i + a sin ωt j)

VELOCITY & ACCELERATION Example 4

Page 36: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Notice that:

F(t) = –mω2r(t)

This shows that the force acts in the direction opposite to the radius vector r(t).

VELOCITY & ACCELERATION Example 4

Page 37: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Therefore, it points toward

the origin.

VELOCITY & ACCELERATION Example 4

Page 38: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Such a force is called a centripetal

(center-seeking) force.

CENTRIPETAL FORCE Example 4

Page 39: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

A projectile is fired with:

Angle of elevation α Initial velocity v0

VELOCITY & ACCELERATION Example 5

Page 40: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Assuming that air resistance is negligible

and the only external force is due to gravity,

find the position function r(t) of the projectile.

VELOCITY & ACCELERATION Example 5

Page 41: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

What value of α maximizes the range

(the horizontal distance traveled)?

VELOCITY & ACCELERATION Example 5

Page 42: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

We set up the axes so that the projectile

starts at the origin.

VELOCITY & ACCELERATION Example 5

Page 43: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

As the force due to gravity acts downward,

we have:

F = ma = –mg j

where g = |a| ≈ 9.8 m/s2.

Therefore, a = –g j

VELOCITY & ACCELERATION Example 5

Page 44: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Since v(t) = a, we have:

v(t) = –gt j + C

where C = v(0) = v0.

Therefore, r’(t) = v(t) = –gt j + v0

VELOCITY & ACCELERATION Example 5

Page 45: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Integrating again, we obtain:

r(t) = –½ gt2 j + t v0 + D

However, D = r(0) = 0

VELOCITY & ACCELERATION Example 5

Page 46: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

So, the position vector of the projectile

is given by:

r(t) = –½gt2 j + t v0

VELOCITY & ACCELERATION E. g. 5—Equation 3

Page 47: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

If we write |v0| = v0 (the initial speed

of the projectile), then

v0 = v0 cos α i + v0 sin α j

Equation 3 becomes:

r(t) = (v0 cos α)t i + [(v0 sin α)t – ½gt2] j

VELOCITY & ACCELERATION Example 5

Page 48: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Therefore, the parametric equations

of the trajectory are:

x = (v0 cos α)t

y = (v0 sin α)t – ½gt2

VELOCITY & ACCELERATION E. g. 5—Equations 4

Page 49: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

If you eliminate t from Equations 4,

you will see that y is a quadratic function

of x.

VELOCITY & ACCELERATION Example 5

Page 50: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

So, the path of the projectile is part

of a parabola.

VELOCITY & ACCELERATION Example 5

Page 51: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

The horizontal distance d is the value

of x when y = 0.

Setting y = 0, we obtain:

t = 0 or t = (2v0 sin α)/g

VELOCITY & ACCELERATION Example 5

Page 52: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

That second value of t then gives:

Clearly, d has its maximum value when sin 2α = 1, that is, α = π/4.

VELOCITY & ACCELERATION

00

2 20 0

2 sin( cos )

(2sin cos ) sin 2

vd x v

g

v v

g g

Example 5

Page 53: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

A projectile is fired with muzzle speed 150 m/s

and angle of elevation 45° from a position

10 m above ground level.

Where does the projectile hit the ground?

With what speed does it do so?

VELOCITY & ACCELERATION Example 6

Page 54: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

If we place the origin at ground level,

the initial position of the projectile is (0, 10).

So, we need to adjust Equations 4 by adding 10 to the expression for y.

VELOCITY & ACCELERATION Example 6

Page 55: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

With v0 = 150 m/s, α = 45°, and g = 9.8 m/s2,

we have:

VELOCITY & ACCELERATION

212

2

150cos( / 4) 75 2

10 150sin( / 4) (9.8)

10 75 2 4.9

x t t

y t t

t t

Example 6

Page 56: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Impact occurs when y = 0, that is,

4.9t2 – 75 t – 10 = 0

Solving this quadratic equation (and using only the positive value of t), we get:

VELOCITY & ACCELERATION

75 2 11,250 19621.74

9.8t

Example 6

2

Page 57: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Then,

x ≈ 75 (21.74)

≈ 2306

So, the projectile hits the ground about 2,306 m away.

VELOCITY & ACCELERATION Example 6

2

Page 58: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

The velocity of the projectile is:

VELOCITY & ACCELERATION

( ) '( )

75 2 (75 2 9.8 )

t t

t

v r

i j

Example 6

Page 59: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

So, its speed at impact is:

VELOCITY & ACCELERATION

2 2| (21.74) | (75 2) (75 2 9.8 21.74)

151m/s

v

Example 6

Page 60: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

When we study the motion of a particle,

it is often useful to resolve the acceleration

into two components:

Tangential (in the direction of the tangent)

Normal (in the direction of the normal)

ACCELERATION—COMPONENTS

Page 61: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

If we write v = |v| for the speed of the particle,

then

Thus, v = vT

ACCELERATION—COMPONENTS

'( ) ( )( )

| '( ) | | ( ) |

t tt

t t v r v v

Tr v

Page 62: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

If we differentiate both sides of that

equation with respect to t, we get:

ACCELERATION—COMPONENTS

' ' 'v v a v T T

Equation 5

Page 63: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

If we use the expression for the curvature

given by Equation 9 in Section 13.3,

we have:

ACCELERATION—COMPONENTS Equation 6

| ' | | ' |so | ' |

| ' |v

v

T TT

r

Page 64: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

The unit normal vector was defined

in Section 13.4 as N = T’/ |T’|

So, Equation 6 gives:

ACCELERATION—COMPONENTS

' | ' | v T T N N

Page 65: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Then, Equation 5 becomes:

ACCELERATION—COMPONENTS

2'v v a T N

Formula/Equation 7

Page 66: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Writing aT and aN for the tangential and

normal components of acceleration,

we have

a = aTT + aNN

where

aT = v’ and aN = Kv2

ACCELERATION—COMPONENTS Equations 8

Page 67: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

This resolution is illustrated

here.

ACCELERATION—COMPONENTS

Page 68: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Let’s look at what Formula 7

says.

ACCELERATION—COMPONENTS

2'v v a T N

Page 69: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

The first thing to notice is that

the binormal vector B is absent.

No matter how an object moves through space, its acceleration always lies in the plane of T and N (the osculating plane).

Recall that T gives the direction of motion and N points in the direction the curve is turning.

ACCELERATION—COMPONENTS

Page 70: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Next, we notice that:

The tangential component of acceleration is v’, the rate of change of speed.

The normal component of acceleration is ĸv2, the curvature times the square of the speed.

ACCELERATION—COMPONENTS

Page 71: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

This makes sense if we think of

a passenger in a car.

A sharp turn in a road means a large value

of the curvature ĸ.

So, the component of the acceleration perpendicular to the motion is large and the passenger is thrown against a car door.

ACCELERATION—COMPONENTS

Page 72: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

High speed around the turn has

the same effect.

In fact, if you double your speed, aN is increased by a factor of 4.

ACCELERATION—COMPONENTS

Page 73: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

We have expressions for the tangential

and normal components of acceleration in

Equations 8.

However, it’s desirable to have expressions

that depend only on r, r’, and r”.

ACCELERATION—COMPONENTS

Page 74: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Thus, we take the dot product of v = vT

with a as given by Equation 7:

v · a = vT · (v’ T + ĸv2N)

= vv’ T · T + ĸv3T · N

= vv’ (Since T · T = 1 and T · N = 0)

ACCELERATION—COMPONENTS

Page 75: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Therefore,

ACCELERATION—COMPONENTS

'

'( ) "( )

| '( ) |

Ta vvt t

t

v a

r r

r

Equation 9

Page 76: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Using the formula for curvature given by

Theorem 10 in Section 13.3, we have:

ACCELERATION—COMPONENTS

2 23

| '( ) "( ) || '( ) |

| '( ) |

| '( ) "( ) |

| '( ) |

N

t ta v t

t

t t

t

r rr

r

r r

r

Equation 10

Page 77: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

A particle moves with position function

r(t) = ‹t2, t2, t3›

Find the tangential and normal components of acceleration.

ACCELERATION—COMPONENTS Example 7

Page 78: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

ACCELERATION—COMPONENTS

2 2 3

2

2 4

( )

'( ) 2 2 3

"( ) 2 2 6

| ( ) | 8 9

t t t t

t t t t

t t

t t t

r = i + j+ k

r = i + j+ k

r = i + j+ k

r'

Example 7

Page 79: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Therefore, Equation 9 gives the tangential

component as:

ACCELERATION—COMPONENTS Example 7

3

2 4

'( ) "( )

| '( ) |

8 18

8 9

T

t ta

t

t t

t t

r r

r

Page 80: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

ACCELERATION—COMPONENTS

2

2 2

'( ) "( ) 2 2 3

2 2 6

6 6

t t t t t

t

t t

ji k

r r

i j

Example 7

Page 81: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Hence, Equation 10 gives the normal

component as:

ACCELERATION—COMPONENTS

2

2 4

'( ) "( )

| '( ) |

6 2

8 9

N

t ta

t

t

t t

r r

r

Example 7

Page 82: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

We now describe one of the great

accomplishments of calculus by showing how

the material of this chapter can be used to

prove Kepler’s laws of planetary motion.

KEPLER’S LAWS OF PLANETARY MOTION

Page 83: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

After 20 years of studying the astronomical

observations of the Danish astronomer

Tycho Brahe, the German mathematician and

astronomer Johannes Kepler (1571–1630)

formulated the following three laws.

KEPLER’S LAWS OF PLANETARY MOTION

Page 84: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

A planet revolves around the sun

in an elliptical orbit with the sun at

one focus.

KEPLER’S FIRST LAW

Page 85: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

The line joining the sun to

a planet sweeps out equal areas

in equal times.

KEPLER’S SECOND LAW

Page 86: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

The square of the period of revolution

of a planet is proportional to the cube

of the length of the major axis of its orbit.

KEPLER’S THIRD LAW

Page 87: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

In his book Principia Mathematica of 1687,

Sir Isaac Newton was able to show that

these three laws are consequences of

two of his own laws:

Second Law of Motion

Law of Universal Gravitation

KEPLER’S LAWS

Page 88: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

In what follows, we prove Kepler’s

First Law.

The remaining laws are proved as exercises (with hints).

KEPLER’S FIRST LAW

Page 89: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

The gravitational force of the sun on a planet

is so much larger than the forces exerted by

other celestial bodies.

Thus, we can safely ignore all bodies in the universe except the sun and one planet revolving about it.

KEPLER’S FIRST LAW—PROOF

Page 90: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

We use a coordinate system with the sun

at the origin.

We let r = r(t) be the position vector

of the planet.

KEPLER’S FIRST LAW—PROOF

Page 91: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Equally well, r could be the position

vector of any of:

The moon

A satellite moving around the earth

A comet moving around a star

KEPLER’S FIRST LAW—PROOF

Page 92: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

The velocity vector is:

v = r’

The acceleration vector is:

a = r”

KEPLER’S FIRST LAW—PROOF

Page 93: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

We use the following laws of Newton.

Second Law of Motion: F = ma

Law of Gravitation:

KEPLER’S FIRST LAW—PROOF

3

2

GMm

rGMm

r

F r

u

Page 94: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

In the two laws,

F is the gravitational force on the planet

m and M are the masses of the planet and the sun

G is the gravitational constant

r = |r|

u = (1/r)r is the unit vector in the direction of r

KEPLER’S FIRST LAW—PROOF

Page 95: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

First, we show that

the planet moves in

one plane.

KEPLER’S FIRST LAW—PROOF

Page 96: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

By equating the expressions for F in

Newton’s two laws, we find that:

So, a is parallel to r.

It follows that r x a = 0.

KEPLER’S FIRST LAW—PROOF

3

GMa

r r

Page 97: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

We use Formula 5 in Theorem 3 in

Section 13.2 to write:

KEPLER’S FIRST LAW—PROOF

( ) ' 'd

dt

r v r v r v

v v r a

0 0

0

Page 98: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Therefore,

r x v = h

where h is a constant vector.

We may assume that h ≠ 0; that is, r and v are not parallel.

KEPLER’S FIRST LAW—PROOF

Page 99: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

This means that the vector r = r(t) is

perpendicular to h for all values of t.

So, the planet always lies in the plane through the origin perpendicular to h.

KEPLER’S FIRST LAW—PROOF

Page 100: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Thus, the orbit of the planet is

a plane curve.

KEPLER’S FIRST LAW—PROOF

Page 101: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

To prove Kepler’s First Law, we rewrite

the vector h as follows:

KEPLER’S FIRST LAW—PROOF

2

2

'

( ) '

( ' ' )

( ') '( )

( ')

r r

r r r

r rr

r

h r v r r

u u

u u u

u u u u

u u

Page 102: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Then,

(Property 6,

Th. 8, Sec. 12.4)

KEPLER’S FIRST LAW—PROOF

22

( ')

( ')

( ') ( ) '

GMr

r

GM

GM

a h u u u

u u u

u u u u u u

Page 103: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

However, u · u = |u|2 = 1

Also, |u(t)| = 1

It follows from Example 4 in Section 13.2 that:

u · u’ = 0

KEPLER’S FIRST LAW—PROOF

Page 104: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Therefore,

Thus,

KEPLER’S FIRST LAW—PROOF

'GM a h u

( ) ' ' 'GM v h v h a h u

Page 105: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Integrating both sides of that equation,

we get:

where c is a constant vector.

KEPLER’S FIRST LAW—PROOF Equation 11

GM v h u c

Page 106: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

At this point, it is convenient to choose the

coordinate axes so that the standard basis

vector k points in the direction of the vector h.

Then, the planet moves in the xy-plane.

KEPLER’S FIRST LAW—PROOF

Page 107: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

As both v x h and u are perpendicular

to h, Equation 11 shows that c lies in

the xy-plane.

KEPLER’S FIRST LAW—PROOF

Page 108: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

This means that we can choose

the x- and y-axes so that the vector i

lies in the direction of c.

KEPLER’S FIRST LAW—PROOF

Page 109: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

If θ is the angle between c and r,

then (r, θ) are polar coordinates of

the planet.

KEPLER’S FIRST LAW—PROOF

Page 110: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

From Equation 11 we have:

where c = |c|.

KEPLER’S FIRST LAW—PROOF

( ) ( )

| || | cos

cos

GM

GM

GMr

GMr rc

r v h r u c

r u r c

u u r c

Page 111: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Then,

where e = c/(GM).

KEPLER’S FIRST LAW—PROOF

( )

cos1 ( )

1 cos

rGM c

GM e

r v h

r v h

Page 112: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

However,

where h = |h|.

KEPLER’S FIRST LAW—PROOF

2

2

( ) ( )

| |

h

r v h r v h

h h

h

Page 113: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Thus,

KEPLER’S FIRST LAW—PROOF

2

2

/( )

1 cos

/

1 cos

h GMr

e

eh c

e

Page 114: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Writing d = h2/c, we obtain:

KEPLER’S FIRST LAW—PROOF

1 cos

edr

e

Equation 12

Page 115: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

Comparing with Theorem 6 in Section 10.6,

we see that Equation 12 is the polar equation

of a conic section with:

Focus at the origin

Eccentricity e

KEPLER’S FIRST LAW—PROOF

Page 116: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

We know that the orbit of a planet is

a closed curve.

Hence, the conic must be an ellipse.

KEPLER’S FIRST LAW—PROOF

Page 117: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

This completes the derivation

of Kepler’s First Law.

KEPLER’S FIRST LAW—PROOF

Page 118: VECTOR FUNCTIONS 13. 13.4 Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal.

The proofs of the three laws show that

the methods of this chapter provide

a powerful tool for describing some

of the laws of nature.

KEPLER’S LAWS