20/07/2006 motion quantities (© f.robilliard) 1flai/theory/lectures/motionquantities.… · si...

20/07/2006 Motion Quantities (© F.Robilliard) 1

vA

A

B

x

y

δt

vB

δv

<a>


The Three Quantities:

.

.

Our aim is to describe the motion of a particle through space. Aparticle is simply a geometrical point. It has no mass. Our description will be geometric, and will have to be made, relative to some assumed reference framework.

We begin by defining suitable quantities to specify the particle’s motion, at any instant. It turns out, that three quantities are sufficient. These quantities are all vectors -

position = location of the particle relative to the specified reference frame

velocity = how quickly the position is changingacceleration = how quickly the velocity is changing

Note: An extended body is treated as a collection of particles


Position:Position is specified by a position vector r. This is an arrow drawn from the origin of coordinates to the particle, at any instant.

Let [position vector of particle P(x,y,z) at time t] = r(t) = r

r = x i + y j + z kwhere x, y, z depend on time t

Say the particle moves along the dotted curve as shown.

SI Units: metre = m

There are two associated quantities – displacement & distance


Displacement:

X

Y A

BrA

rB

δrSay a particle moves, along the dotted curve, from a point A to a point B. Say rA = position vector of point A

rB = position vector of point B

The displacement of the particle from A to Bis the straight arrow (vector) AB.

Displacement = AB = change in position = δr = rB -rA

By vector addition ( rA + δr ) = rB

Since AB is the change in position vector r, it is represented by δr.

SI Unit: metre = m

(final position vector) - (initial position vector)

Transposing gives δr = ( rB – rA)


Distance = Path Length:Say a particle moves, along the dotted path, from a point A to a point B.

The Distance, d, moved by the particle is the total scalar path length along theactual path travelled by the particle.

In general, the path will be curved.

Since vectors are straight, path length (distance) must be a scalar!

SI Units: metre = m

X

Y A

BrA

rB

d


Instantaneous Velocity, v:

Say a particle moves along the dotted curve shown.

rA

rB

δrA

B

x

y

δtSay it takes a time δt, to move from point Aon the curve, to a nearby point B.

rA = position vector of ArB = position vector of B

The change in position vector from A to B = rB - rA = δr

If we take increasingly short time intervals δt, B moves closer to A, vector δr gets shorter, its direction approaches that of a tangent at A,and the magnitude of ratio (δr)/(δt) approaches the instantaneous rate at which the particle A moves along its path, per unit time.

Velocity is the time-rate of change of position.


Definition of v

dtd

�t�

0�t

limit

takeninterval timesmall ingcorrespondAnear particle ofector position vin change

Aat particle ofposition of change of rate-time

Aat particle of velocityousinstantane

r

r

v

≡

��

��

→≡

��

��

≡

��

��

≡

��

��

≡

dtdrv ≡

SI Units: m/s = m s-1

A

BrArB

δr

x

y

δt

v


v in Cartesian Cordinates:

( )

.directions z & y, x,in the

components velocity theare v, v, vwhere

kvjviv

kdtdz

jdtdy

idtdx

k z jy ix dtddtdr

then v

t timeof functions are z & y, x, wherek, z jy i x rGiven

zyx

zyx ++=

++=

++=

≡

++=

dtdx

v:dimension 1In x =


Other Associated Quantities:Instantaneous Speed is the magnitude of the instantaneous velocity.If the velocity is v, then the speed is | v | = v.

Average Velocity is the total displacement vector divided by the total time taken.

. r,rvv large bemay �ly consequent and �t, where, �t�==

Average Speed is the total scalar pathlengthdivided by the total time taken


Relative Velocity:It is a very fundamental principle in Physics, that velocity onlyhas meaning when measured relative to some reference framework.

In this sense, all velocities are relative, and the value measuredwill depend on the particular measuring framework.

y 5 m/s 5 m/s

A B x

The labeled velocities in the figures below are those measured by the xy frame.

5 m/s 5 m/s

A B x

y

The xy frame sees B moving with5 m/s in the (+x) direction

A sees B to be stationary.

The xy frame sees B moving with5 m/s in the (-x) direction.

A sees B to be approaching, in the (-x) direction, with a velocity of 10 m/s.


3 Dimensional Case:

xz

y

vB

vAvA = velocity of particle A relative to xyz axes

vB = velocity of particle B relative to xyz axes

The velocity of A relative to B is the velocitythat A would be measured to have by B = vAB

To find this, we add a velocity (-vB) to both A and B.

This brings B to rest, and gives A its velocity relative to B(It is equivalent to jumping aboard B).

vBvAB = vA - vB = ( vA ) - ( )

= ( vA ) + ( (-vB) )vAB

(-vB)

vA

velocity ofA relative to B


Example:Rain falls vertically downward at 8 m/s.A cyclist rides along a horizontal road at 6 m/s.Find the velocity of the rain as measured by the cyclist.

x

y

vRvC

vR = velocity of Rain = - 8 j m/svC = velocity of Cyclist = +6 i m/s

(velocity of rain relative to cyclist )= vRC= vR - vC= (-8 j) - (6 i)= (-6 i -8 j ) m/s

Magnitude:|vRC| = (62 +82)1/2

= 10 m/s

Direction:tan θ = 8/6θ = 53.1 deg.

vR-vC

vRC

θ

xy

86

Rain hits cyclist diagonally in the face.


Instantaneous Acceleration, a:If a particle changes its velocity, in either magnitude or direction,it has accelerated.

Its acceleration is the time-rate of change of its velocity vector.

Say a particle moves along the dotted curve, as shown.

vA

vBvA is the velocity vector of particle at A

It moves from a point A on the curve, to a nearby point B, in a time interval of δt

A

B

x

y

δt

vB is the velocity vector of particle at B

(Remember: a velocity will have the direction of the tangent to the curve.)


Change in Velocity:The change in velocity vector, δv, is the vector that must be added tothe initial velocity, vA, to equal the final velocity, vB.

Change = final - originalδv = vB - vA

vAA

B

x

y

δt

vBvB= ( ) – ( )

vA

vB= ( ) + ( )vA

vB

vA

δv

δv


Instantaneous Acceleration:

( )( )

�t�v

takentimein change

��

��≡

≡��

�

�

a

velocityonaccelerati

average

dtd

�t�

0�tlimit

onacceleratiousinstantane vva ≡

��

��

→≡≡��

�

�

�

vAA

B

x

y

δt

vB

δv

<a>

To get the instantaneous acceleration,we reduce δt. B moves up the curve toward A, δv is correspondingly reduced, and the ratio (δv/δt) approaches a limiting value.

vAAB

x

y

vBδv

a

0v →�

Unit: m s-2


In Cartesian Coordinates:

( )

.directions z & y, x,in the

componentson accelerati theare a ,a ,a where

kajaia

kdt

dvj

dt

dvi

dt vd

k v j v i vdtddtdv

athen

t timeof functions are v& , v, v wherek, v j v i vGiven v

zyx

zyx

zyx

zyx

zyxzyx

++=

++=

++=

≡

++=

dtdv

a :dimension 1In xx =


Summary:

(r)dtd

(v)dtd

a and (r)dtd

v 2

2

≡≡≡

Position, r, velocity, v, and acceleration, a, are connected by definition.

These can be expressed inversely:

=→≡=→≡ a.dtvdtdv

a and v.dtrdtdr

v

If any one of a, v, or r is known, the other two can be determined, using the above relationships.

In fact, the general problem of motion is – given one of a, v, r, find the other two.

20/07/2006 motion quantities (© f.robilliard) 1flai/theory/lectures/motionquantities.… · si...

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