Ch. 22 Vectors and Relative Velocity

8/2/2019 Ch. 22 Vectors and Relative Velocity

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Singapore International School (Hong Kong)/ S4 Add Math (2010)

1

Vectors and Relative Velocity

1. Notations to represent unit vectors along the x-axis and y-axis

There are two notations to represent unit vectors in the positive direction of thex-axis (eastbound) and

y-axis (northbound):

Description Notation: i-j Notation: Column

vector

Eastbound vector of length 1 i 1

0

!

"#$

%&

Northbound vector of length 1 j 0

1

!

"#$

%&

Example 1.1

(a) The vector 2i + 3j represents a vector travelling 2 units east and 3 units north.

(b) The vector 2i 3j represents a vector travelling 2 units east and 3 units south.

(c) The vector!2

!3

"

#$%

&'represents a vector travelling 2 units west and 3 units south.

Result 1.2: To calculate the bearing of a vector written in i-j or column vector notation,

we first represent the vector on a diagram and apply trigonometry a right-angled triangle.

Exercise 1.3Represent the following vectors on the same diagram. Hence or otherwise, calculate the bearing ofthese vectors:

(a) 2i + 2j

(b)1

!2

"

#$%

&'

(c) !2i ! 3 j

(d)2

! 3"#$ %

&'


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2

2. Magnitude and Converting vectors to unit vectors

The magnitude or length of vector v = ab

!

"#$

%&is (via Pythagoras Theorem):

v = a2 + b2.

Recall: A unit vector is a vector with magnitude 1 unit.

All non-zero vectors can be converted to a unit vector, as shown in the example below:

Example 2.1(a) Calculate the length of the vectorv = 3i + 4j.

(b) Hence write down a unit vector in the same direction as v.

Result 2.2 : Formula to find a unit vector ! in the same direction as v is

!=

1v vv .

Exercise 2.3

Given that a = 3i 4j, find the vectors b and c given that:

(a) Vectorb is a unit vector in the same direction as a (b) Vectorc is a vector of magnitude 10 in the opposite direction ofa.


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Result 2.4 : Given the bearing of a vector and its magnitude, we can find the vector in i- j or column

vector notation, by first drawing a right-angled triangle and applying trigonometry.

Exercise 2.5Express the following vectors in the form ai + bj:

(a) Vectorp, which has a bearing of 030 and a magnitude of 10

(b) Vectorq, which has a bearing of120

and a magnitude of 15(c) Vectorr, which is travelling southwest and has a magnitude of 18

(d) Vectors, which is travelling northwest and has a magnitude of 20.

3. Vectors, Position, and Distances

We can represent the position of particles using vectors, and calculate distances between them.

Example 3.1

The position vectors of particlesA and B are given by1

7

!

"#$

%&and

!2

!5

"

#$%

&'respectively.

(a) Find the vectorAB! "!!

.

(b) Hence find the distance betweenA and B.


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Result 3.2 : To find the distance betweenA and B, first find AB! "!!

and then find AB! "!!

.

Exercise 3.3To an observer at position 2i 3j, a boat appears to be 5i + 9j units away.

(a) Find the distance between the observer and the boat.(b) Find the position of the boat.

4. Vectors, Velocity and Spee ds

Definition 4.1 : Speed is the magnitude of velocity, i.e. if velocity v = ab

!

"#$

%&, then:

Speed, v = a2 + b2

Recall The velocity ofA relative to B is given by vA/B

= vA! v

B

Example 4.2

A boy jogs with velocity vector represented by3

4

!

"#$

%&while a girl jogs with velocity vector represented by

!1

!3

"

#$%

&'.

(a) Who is jogging at a faster speed, and by how much?

(b) Find the relative velocity of the boy relative to the girl.(c) Hence find the speed of the boy relative to the girl, and the bearing of this relative velocity.


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Result 4.3 : If a particle is traveling in the same direction asa

b

!

"#$

%&, then its velocity vector is given by

ka

b

!

"#

$

%&

, where kis a constant to be found.

Exercise 4.4

A ship travels in the direction i + 9j. Given that its speed is 20 km/h, find the velocity vector of the ship.

5. Vectors, Displacement and Velocities

Result 5.1 : Suppose a particle has displacement given by s = ab

!

"#$

%&+ t

c

d

!

"#$

%&at time t, where a, b, cand

dare constants. Then:

(a) The particles initialdisplacement is ata

b

!

"#

$

%&

(b) The particles velocity is v = ddt

s( ) =c

d

!

"#$

%&

(c) The particles speed is given by v = c2 + d2

Exercise 5.2A particle has displacement given by s = (1 4t)i + (2 + 3t)j.

(a) Find its initial distance away from the origin.

(b) Find the direction in which it is travelling and its speed.


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Exercise 5.3

A particle travels in the direction 3i 4j with a speed of 15 cm/s.

(a) Find the velocity vector of the particle.(b) Given that its initial position is 2i + 4j, find its displacement at time t.


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6. Vectors and Relative Velocity

Result 6.1

Two methods to find the time for a particle to collide (or intercept) another particle:

(a) Method 1: Use the fact that both positions at time tare equal at time of collision(b) Method 2: Use the fact that position ofB relative toA is 0 at time at collision, i.e.

t vB/A( ) = BA

! "!!

( )initial

Exercise 6.1

ObjectA is at position!8 + 4t

3 + 3t

"

#$%

&'and object B at

2 + 2t

!7 + 5t

"

#$%

&', where tis time measured in minutes. All

distances are measured in metres.(a) Find the initial distance betweenA and B.(b) Find the time whenA and B are 10 metres apart.

(c) Find the velocity ofB relative toA.

(d) Determine ifA and B will collide, and if yes, the time of collision.


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Exercise 6.2

A plane flies fromA to B. The velocity in still air of the plane is (250i 40j) km/h and there is a

constant wind blowing with velocity (40i 60j) km/h. Find:(a) The bearing ofB fromA(b) The time of flight, given that the distance AB is 250 km.

Exercise 6.3

A plane flies fromA to B, where PQ! "!!

=

900

300

!

"#$

%&km . A constant wind blows northwest with speed

50kmh!1 .

(a) Find the velocity of the wind, giving your answer in the forma

b

!

"#$

%&kmh

'1.

Given that the plane takes 3.5 hours to travel fromA to B, find:

(b) The velocity, in still air, of the plane, giving your answer in the forma

b

!

"#

$

%&

kmh'1 ,

(c) The bearing, to the nearest degree, on which the plane must be directed.


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Relative Velocity

True Velocity and Relative Velocity

Consider an athleteA jogging along a straight road at a speed of due east.

We can represent the velocity of athleteA as . This velocity is relative to the earth and is known as the true

(oractual ) velocity .

Similarly is the true velocity of a cyclist who travels at a speed of due west.

The velocity of a moving objectPrelative to the earth is known as the true (oractual ) velocity and is denoted

by .

Suppose thatPand Q are two fixed points on the earth and athlete A jogs fromPto Q in 30 minutes. The true

distance fromPto Q is:

Relative v elocity

Consider two menPand Q running due east at velocities of and respectively. Q is initially aheadofP.

Intuitively, we observe that:

To the man Q, the man Pappears to move at a speed of towards man Q, i.e. due east. We call

this the velocity ofP relative to Q, . This velocity is also known as a relative vel ocity.

We can use relative velocity to find the distance between the two running men. Suppose man Q isinitially 20 m due east of man P. Observe that is parallel to . Then the time that man Pneeds

to overtake man Q is:

We also observe that:

2ms!1

vA

vC

20 kmh!1

v

P

PQ= v

At= 2 30 ! 60( ) =3 600 m

3 ms

!1

2 ms

!1

1ms

!1

vP/Q

vP/Q PQ

! "!!

Initial distance between PandQ

vP/QP/Q

=

20

1= 20 s


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To the manP, the man Q appears at a speed of due west towardsP. We call this the velocity ofQ

relative toP, .

The velocity of a moving object Prelative to another moving object Q is known as apparent velocity

and is denoted .

If is parallel to , then:

Note

is a vector quantity, i.e. it has magnitude and direction.Addition and subtraction of vectors is done using the triangle or parallelogram law.

Example 1.1

A and B are two fixed points on a straight path and the distance apart is 30m. A man Pjogs from A to B at a

speed of . Another man Q jogs fromB at a speed of in the opposite direction. Find

(a) the velocity ofPrelative to Q,

(b) the time taken forPand Q to pass each other.

Solution

(a) Using , velocity ofPrelative to Q .

(b) Since is parallel to , time taken

1 ms!1

vQ/P

vP/Q

vP/Q

= vP! v

Q

vP/Q

= !vQ/P

vP/Q PQ

! "!!

Time taken forPand Qto meet =Initial distance between Pand Q

vP/Q

vP/Q

3 ms

!1

2 ms

!1

vP/Q

= vP! v

Q= 3 ! !2( ) = 5 ms!1

vP/Q PQ

! "!!

=

30

5= 6 s

P


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Exercise 1.2

A straight horizontal moving walkway travels at in a direction from a fixed point A towards another

fixed pointB. A man on the moving walkway is walking at a speed of fromA toB. Write down

(a) the velocity of the man relative to the walkway,

(b) the true velocity of the man.

Solution

(a) Velocity of man relative to walkway, , in the direction fromA to B.

(b) Since , true velocity of man

Example 1.3

A river is flowing at a speed of due east. A boat, whose speed in still water is , is moving

upstream. Find the true speed and the actual direction of motion of the boat.

Solution

Note that velocity of boat relative to the water is the same as its velocity in still water.

Since , .

Then, true speed of boat is and direction of motion is due west.

Exercise 1.4

1.5 ms!1

1ms

!1

vM/W

= 1 ms!1

vM/W

= vM! v

W = v

M/W+ v

W= 1+1.5 = 2.5ms

!1

3 ms

!1

5 ms

!1

vB/W

= vB! v

W vB= v

B/W+ v

W

2 ms

!1


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In an airport, a straight horizontal moving walkway travels at i the direction from the fixed point A

towards the fixed point B. A passengerPwalks from A to B on the moving walkway at a speed of

relative to the walkway. At the same instance, another passenger Q walks from B to A on a fixed horizontal

ground alongside the walkway, at a speed of . Calculate

(a) the velocity ofP,

(b) the speed ofPrelative to Q.The distanceA B is 120 m. SupposePand Q pass each other aftertseconds, find the value oft.

Solution

Practice Set 1

0.8 ms!1

1ms

!1

1.2 ms

!1


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1. A andB are two fixed points on a straight path and the distance apart is 110 m. A boyPruns fromA at a

speed of towards the pointB. A girl Q runs fromB at a speed of in the opposite direction.

Find

(a) the velocity ofPrelative to Q,

(b) the time taken forPto meet Q and the distance travelled byP.

2. A straight horizontal moving walkway travels at in a direction from the fixed pointA towards the

fixed pointB. A manPwalks fromA toB on the moving walkway at a speed of . Write down

(a) the velocity of the man relative to the walkway,

(b) the true velocity of the man.

3. A river is flowing at a speed of due east. A boat, whose speed in still water is , is moving

upstream. Find the true speed and direction of motion of the boat. At a certain instant, the river is flowing

at a slower speed of . Find the true speed of the boat at that instant.

4. In an international airport, a straight horizontal moving walkway is designed to travel at in a

direction from the fixed point A towards the fixed point B. A passengerPwalks from A to B on themoving walkway at a speed of relative to the walkway. At the same instant, another passengerQ

walks fromA toB, on a fixed horizontal ground alongside the walkway at a speed of . Calculate

(a) the velocity ofP,

(b) the speed ofPrelative to Q.

The distanceAB is 120 m. Find the distance betweenPand Q at the instant whenPreachesB.

5. A straight horizontal moving walkway travels at in a direction from the fixed point A towards the

fixed point B. A man P walks from A to B on the moving walkway at a speed of relative to the

walkway. Another man Q walks from B to A, on a fixed horizontal ground alongside the walkway, at a

speed of . Calculate the speed of P relative to Q.The distanceA B is 200 m andPand Q pass each other at a point halfway between A and B. Find the time

between Qs departure fromB andPs departure fromA .

6. A carA and a van B are moving in the same direction on a straight road. CarA is travelling at

and vanB is travelling at . Write down the velocity ofB relative toA . Hence find the time taken

from the instant whenA andB are at a distance 200 m apart to the instant whenA catches up withB.

Answe rs to Practice S et 1

1. (a) (b) 20 s, 40 m (???)

2. (a) (b)

3. upstream, downstream (???)

4. (a) (b) , 30 m

5. , 33.9 s

6. , 50 s

2.5 ms!1

3 ms

!1

1.2ms!1

1.3ms

!1

3 ms

!1

5 ms

!1

2 ms

!1

0.8 ms

!1

1.2ms

!1

1.5 ms

!1

2 ms

!1

1.5 ms

!1

1.6ms

!1

12ms

!1

8 ms

!1

vP/Q

= 5.5 ms!1

1.3ms!1

2.5 ms!1

2 ms

!1

3 ms

!1

2 ms

!1

0.5 ms

!1

5.1 ms

!1

4 ms

!1


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Relative Motion in a Current

River crossing problems: A pply triangle or parallel ogram law if necess ary

Example 2.1

A river is flowing at due east. A boat, of speed in still water, is steered in the direction due north.

Find the true speed and direction of motion of the boat.

Solution

LetB denote the boat and W the water.

east and north.

Using , we find that:

Angle between boat and stream is .

Thus, the true speed of the boat is and it travels upstream making an angle with the bank.

For an objectPmoving in the water, the course taken by Pis the direction of , the velocity ofPrelative to

W in the water.

Perpendicular components of a vel oci ty

In the above example, can be resolved into two perpendicular components, one in the horizontal

direction (which is in the direction of the current) and the other in the vertical direction (which enables the boat to

cross the river).

3 ms!1

4 ms

!1

vW= 3 ms

!1

vB/W

= 4ms!1

vB= v

B/W+ v

W

v

B= 3

2+ 4

2= 5 ms

!1

tan!1 4

3

"

#$%

&'= 53.1

5 ms

!1

53.1

vP/W

v

B= 5 ms

!1


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Graphical method

In the above

examples, we use the calculation method to find the unknown speeds and directions

with the aid of a rough sketch for each velocity diagram.

However, if the question allows (or

requires), we can construct a velocity diagram to

scale on graph paper and measure the unknown speeds and directions. This method is more convenient

than the

calculation method

for problems involving more computations.

Motion of aircraft in the air

Example 2.4

The speed of a commercial aircraft in still air is . The wind velocity is from the west. The

aircraft is steered on course in the direction . Find the true velocity of the aircraft.

SolutionLetA denote the aircraft and W the wind.

and

Apply to draw a velocity diagram as shown:

Using Cosine Rule, true speed is:

Using Sine Rule:

The true velocity of the aircraft is and the bearing is .

300 kmh!1

80 kmh

!1

060

vA/W

= 300 kmh!1

vW= 80kmh

!1

vA= v

A/W+ v

W

vA

= 3002+ 80

2! 2 300( ) 80( )cos150

= 371 kmh!1

sin ! " 60( )80

=sin 150( )

371

! " 60 = 6.2

! = 66.2

371kmh

!1

066.2

3.6ms-1

B

3ms-1

4

v

3

2 80 m

60 m

4 ms-1


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For any objectPmoving in the air

the speed ofPin still air is

the course taken byPis the direction of , where is the velocity ofPrelative to the wind.

Exercise 2.5

An aircraft flies due north from A to B, where AB = 252 km. The wind is blowing from the direction at

. The speed of the aircraft in still air is and the pilot sets the course on the bearing ,

where is acute. Find

(a) the value of

(b) the time taken in minutes for the journey fromA toB.

Solution

Exercise 2.6

(a) In a wind blowing at from the direction , a cyclist travels due west at . Find the

velocity of the wind relative to the cyclist.

(b) A boat travels due north at a speed of . To the man in the boat, the wind appears to blow at

from the bearing . Find the actual velocity of the wind.

Solution

Practice Set 2

1. A river is flowing at due south. A boat, whose speed in still water is , is steered in the

direction due east. Find the true speed and direction of the motion of the boat.

2. A river is flowing at due east. A speedboat, whose speed in still water is , is steered in the

direction on a bearing of . Find the resultant velocity of the speedboat.

3. A river is flowing at a speed of due east. A boat, whose speed in still water is , is steered

in the direction due north. Find the true velocity of the boat.

vP/W

vP/W vP/W

040

85kmh

!1

350 kmh

!1!

!

!

3 ms

!1

060 4.5 ms

!1

20kmh

!1

10 kmh

!1

030

4 ms

!1

3 ms

!1

3 ms

!1

5 ms

!1

330

2.5ms

!1

6 ms

!1


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4. A soldier who swims at in still water wishes to cross a river 20 m wide. The water is flowing

between straight parallel banks at . He swims upstream in a direction making an angle of

with the bank. Find

(a) the resultant velocity

(b) the time taken for the crossing, to the nearest second.

5. The speed of an aircraft in still air is . The wind velocity is from the east. The

aircraft is steered on the course in the direction . Find the true velocity of the aircraft.

6. An aircraft flies due east fromA toB,whereAB = 200 km. The wind is blowing from the direction

at . The speed of the aircraft in still air is and the pilot sets the course on the bearing

. Find

(a) the value of

(b) the time taken, in minutes, for the journey fromA toB.

7. An aircraft is flying due south at . The wind is blowing at from the direction ,

where is acute. Given that the pilot is steering the aircraft in the direction , find(a) the value of ,

(b) the speed of the aircraft in still air.

8. A man who swims at in still water wishes to cross a river which is flowing between straight

parallel banks at . He aims downstream in a direction making an angle with the bank. Find

(a) the speed at which he travels,

(b) the angle which his resultant velocity makes with the bank.

Answe rs to Practice S et 2

1. in the direction

2. on the bearing

3. , makes an angle (downstream) with the bank

4. (a) , makes an angle (downstream) with the bank (b) 18 s

5. on the bearing

6.(???)

7. (a) (b)(???)

8. (a) (b)

Relative Motion of Two Moving Objects

1.2 ms!1

1.8 ms

!1

70

300 kmh!

1

60 kmh!

1

060

030

60 kmh

!1

300 kmh

!1

!

!

350 kmh

!1

70 kmh

!1!

! 170!

!

1.2ms

!1

2 ms

!1

60

5 ms

!1

143.1

4.36ms

!1

006.6

6.5ms

!1

67.4

1.8ms

!1

40

250 kmh

!1

053.1

! = 300, 48 s

50.3 313 kmh

!1

2.8 ms

!1

21.8


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For two objects Pand Q moving with velocity and respectively, we can apply the same relative velocity

equation:

Apparent (relative) path

Example 3.1

At a particular instant, two shipsPand Q are 5 km apart and move with constant speeds and directions as shown.

Find

(a) the speed and direction ofPrelative to Q,

(b) the distance apart, in metres, whenPis due south ofQ.

Solution

(a) Using :

To find direction of this relative velocity:

Thus, motion ofPrelative to Q is in the direction .

(b) The path of the motion ofPrelative to Q (observer) is shown in the diagram below:

When the shipPis due south ofQ, the distance apart is:

In the above example, the path ofPobserved by Q is known as the apparent path and the velocity is in the

direction of the motion ofPalong the path.

Interception

vP

vQ

vP= v

P/Q+ v

Q

vP= v

P/Q+ v

Q

vP/Q

= vP! v

Q

vP/Q

= 82+10

2= 12.8 kmh

!1

! = tan"1 8

10

#

$%&

'(= 38.7

060 +! = 098.7

d= 5tan 8.7( ) = 765 m

vP/Q

B A

8

300kmh-1

30o

60o5 km

10 kmh-1

P


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Example 3.2

At an instant, two cyclists A and B are 120 m apart with B due east ofA . CyclistA is travelling at in a

direction and cyclistB is travelling at in a direction . Find

(a) the value of for which the cyclistB should travel in order to intercept cyclist A ,

(b) the time taken for the interception to occur.

Assuming that cyclist A is travelling at a speed of in the same direction and the speed of cyclist B

remains unchanged, show that there are two possible directions in which B should travel in order to intercept A .

Indicate the direction in whichB should travel in order to interceptA as quickly as possible.

Solution

(a) For interception,B appears to move towardsA as shown:

We draw a velocity diagram as shown:

Thus,

(b) From the velocity diagram

Time taken is .

Applying with , we can draw the velocity diagram as shown:

3 ms!1

030 3.8 ms

!1!

!

4 ms!1

sin!

3=sin60

3.8

sin! = 0.684

!" 43.1

! = 270 + " = 313.1

vB/A

sin76.9=

3.8

sin60

vB/A

= 4.274ms!1

120

4.274= 28.1s

vB= v

A+ v

B/A vA= 4ms

!1

Q

8 kmh-1

P

Q

8.7

vPQ

Path of P60

o


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For two objects P and Q moving in a plane, it is important to obtain the following information:

, the initial position vector of P relative to Q , the velocity of P relative to Q.

and are in opposite directions

To Q (observer),Pappears to move towards QPwill intercept Q.

Practice Set 3

1. Two particlesPand Q are 30 m apart with Q due north ofP. Particle Q is moving at in a direction

and P is moving at in a direction . Find

(a) the magnitude and direction of the velocity of Q relative to P,

(b) the time taken for Q to be due east of P, to the nearest second.

2. At a particular instant, two boatsPand Q are 2 km apart andPis due north ofQ. Pmoves with constant

speed due east while Q moves with constant speed in direction . Find

(a) the speed and direction ofQ relative toP,

(b) the distance apart in metres when Q is due west ofP.

3. At a particular moment, two shipsA and B are 5 km apart with A due west ofB. ShipA is sailing due

south at and shipB is sailing due west at . Find

(a) the velocity ofA relative toB,

(b) the distance between the two ships whenA is on the bearing of fromB.

4. Two aircraft A and B fly at the same height with constant velocities. At noon, aircraft B is 50 km due

east of aircraft A and is flying due west at . Aircraft A is flying on the bearing at

. Find

(a) the velocity ofB relative toA ,

(b) the time whenB is due north ofA .

5. Particle Q is initially 50 km east of particle P. Particle Pmoves with speed at direction

while particle Q moves with speed at direction . Given thatPand Q are on the path

of collision, find

(a) the value of ,

(b) the time that elapses before the collision, to the nearest second.

6. At a given instant, an airship is moving due north with a speed of . A helicopter which is 500 m

due east of the airship flies at a speed of and steers on a bearing of in order to intercept the

airship. Find

(a) the value of ,

(b) the time that elapses before the interception.

QP

! "!!

vP/Q

vP/Q QP

! "!!

!

!

5 ms

!1

090 7 ms

!1

030

3 ms

!1

4 ms

!1

060

5 kmh

!1

8 kmh

!1

225

450 kmh

!1

120

300 kmh

!1

8 ms

!1

060

10 ms

!1

270 + !( )

!

6 ms

!1

15 ms

!1!

!


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Ch. 22 Vectors and Relative Velocity

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