Motion in One Dimension Pt 1

7/30/2019 Motion in One Dimension Pt 1

1/4

PositionAny object is situated at point O and three

observers fromthree differentplaces arelooking atsame object,then all threeobservers willhave differentobservations

about the position of point O and no one will be wrong.Because they are observing the object from differentpositions.

Observer A says : Point O is 3 m away in westdirection.

Observer B says : Point O is 4 m away in southdirection.

Observer C says : Point O is 5 m away in eastdirection.

Therefore position of any point is completelyexpressed by two factors: Its distance from the observerand its direction with respect to observer.

That is why position is characterised by a vectorknown as position vector.

Consider a point P in xy planeand its coordinates are ( x , y ). Thenposition vector )( r of point will be

j y i x

+ and if the point P is in spaceand its coordinates are ( x , y , z ) thenposition vector can be expressed as . k z j y i x r ++=

Rest and Motion

If a body does not change its position as timepasses with respect to frame of reference, it is said to

be at rest.And if a body changes its position as time passes

with respect to frame of reference, it is said to be inmotion.

Frame of Reference : It is a system to which a set of coordinates are attached and with reference to whichobserver describes any event.

A passenger standing on platform observes that atree on a platform is at rest. But the same passengerpassing away in a train through station, observes thattree is in motion. In both conditions observer is right.

But observations are different because in first situationobserver stands on a platform, which is reference frame

at rest and in second situation observer moving in train,which is reference frame in motion.

So rest and motion are relative terms. It dependsupon the frame of references.

Table 2.1 : Types of motion

Onedimensional

Twodimensional

Threedimensional

Motion of a bodyin a straight lineis called onedimensionalmotion.

Motion of body ina plane is calledtwo dimensionalmotion.

Motion of body ina space is calledthree dimensionalmotion.

When only onecoordinate of theposition of abody changeswith time then itis said to bemoving one

dimensionally.

When twocoordinates of the position of abody changeswith time then itis said to bemoving two

dimensionally.

When all threecoordinates of theposition of a bodychanges with timethen it is said tobe moving threedimensionally.

Ex. . (i) Motion of car on a straightroad.

(ii) Motion of freely fallingbody.

Ex. (i) Motion of car on a circularturn.

(ii) Motion of billiards ball.

Ex. . (i) Motion of flying kite.

(ii) Motion of flying insect.

Particle or Point Mass or Point object The smallest part of matter with zero dimension

which can be described by its mass and position isdefined as a particle or point mass.

If the size of a body is negligible in comparison to itsrange of motion then that body is known as a particle.

A body (Group of particles) can be treated as aparticle, depends upon types of motion. For example ina planetary motion around the sun the different planetscan be presumed to be the particles.

In above consideration when we treat body asparticle, all parts of the body undergo samedisplacement and have same velocity and acceleration.

Distance and Displacement(1) Distance : It is the actual length of the path

covered by a moving particle in a given interval of time.

Motion in one Dimension

C4m3

m

EW

N

5m

O

A

BS

Fig. 2.1

Y

X

Z

r

P( x,y,z )


2/4

(i) If a particle starts from A and reach to C throughpoint B as shown in the figure.

Then distance travelled by particle

7=+= BC AB m

(ii) Distance is a scalar quantity.(iii) Dimension : [ M0L1T 0]

(iv) Unit : metre (S.I.)

(2) Displacement : Displacement is the change inposition vector i.e., A vector joining initial to finalposition.

(i) Displacement is a vector quantity

(ii) Dimension : [ M0L1T 0]

(iii) Unit : metre (S.I.)(iv) In the above figure the displacement of the

particle BC AB AC += || AC

oBC ABBC AB 90cos)()(2)()( 22 ++= =

5 m

(v) If nSSSS ........,, 321 are the displacements

of a body then the total (net) displacement is the vector

sum of the individuals. nSSSSS ++++= ........321

(3) Comparison between distance anddisplacement :

(i) The magnitude of displacement is equal tominimum possible distance between two positions.

So distance |Displacement|.(ii) For a moving particle distance can never be

negative or zero while displacement can be.(zero displacement means that body after motion

has came back to initial position)i.e. , Distance > 0 but Displacement > = or < 0(iii) For motion between two points, displacement is

single valued while distance depends on actual pathand so can have many values.

(iv) For a moving particle distance can neverdecrease with time while displacement can. Decrease indisplacement with time means body is moving towardsthe initial position.

(v) In general, magnitude of displacement is notequal to distance. However, it can be so if the motion isalong a straight line without change in direction.

(vi) If Ar and Br are the position vectors of particle initially and finally.

Then displacement of the

particle AB AB r r r =

and s is the distancetravelled if the particle hasgone through the path APB .

Speed and Velocity(1) Speed : The rate of distance covered with time is

called speed.

(i) It is a scalar quantity having symbol .

(ii) Dimension : [ M0

L1

T 1

](iii) Unit : metre/second (S.I.), cm /second (C.G.S.)(iv) Types of speed :

(a) Uniform speed : When a particle coversequal distances in equal intervals of time, (nomatter how small the intervals are)

(b) then it is said to be moving with uniformspeed. In given illustration motorcyclisttravels equal distance (= 5 m ) in eachsecond. So we can say that particle is movingwith uniform speed of 5 m /s .

(b) Non-uniform (variable) speed : In non-uniform speed particle covers unequal distances inequal intervals of time. In the given illustrationmotorcyclist travels 5 m in 1 st second, 8 m in 2 nd second,10 m in 3 rd second, 4 m in 4 th second etc .

Therefore its speed is different for every timeinterval of one second. This means particle is movingwith variable speed.

(c) Average speed : The average speed of a

particle for a given Interval of time is defined as theratio of total distance travelled to the time taken.

Average speedtaken Time

travelledistance Total= ;

t

sv av

=

4m

3 m B

C

A

Fig. 2.2

Time

Uniform Speed

5 m 5 m

1 sec 1 sec 1 sec 1 sec 1 sec

5 m 5 m 5m

5m/s

5m/s

5m/s 5m/s

5m/s

5 m

5 m/s

1 m/s

Fig. 2.4

Distance

Time

Variable Speed

5m 8 m

1 sec 1 sec 1 sec 1 sec 1 sec

10m

4 m 6 m

5m /s

8 m /s

10 m /s

4 m /s

6 m /s

7m

1 sec

7 m /s

Fig. 2.5

Y

X

A

B

ABr Br

Ar

s

Fig. 2.3

P


3/4

Time average speed : When particle moves with

different uniform speed 1 , 2 , 3 ... etc in different

time intervals 1t , 2t , 3t , ... etc respectively, itsaverage speed over the total time of journey is given as

elapsed Total timecoveredistance Total=av v

......

......

321

321

+++

+++=

t t t

d d d =

......

......

321

332211

+++

+++

t t t

t t t

Distance averaged speed : When a particle

describes different distances 1d , 2d , 3d , ...... with

different time intervals 1t , 2t , 3t , ...... with speeds

......,, 321 v v v respectively then the speed of particleaveraged over the total distance can be given as

elapsed Total timecoveredistance Total

=av

......

......

321

321

+++

+++=

t t t

d d d

......

......

3

3

2

2

1

1

321

+++

+++=

d d d

d d d

If speed is continuously changing with time then

=

dt

vdt v av

(d) Instantaneous speed : It is the speed of aparticle at a particular instant of time. When we sayspeed, it usually means instantaneous speed.

The instantaneous speed is average speed forinfinitesimally small time interval ( i.e ., 0 t ). Thus

Instantaneous speedt

sv

t =

0lim

dt

ds=

(2) Velocity : The rate of change of position i.e.rate of displacement with time is called velocity.

(i) It is a vector quantity having symbol v .(ii) Dimension : [ M0L1T 1](iii) Unit : metre/second (S.I.), cm /second (C.G.S.)(iv) Types of velocity :(a) Uniform velocity : A particle is said to have

uniform velocity, if magnitudes as well as direction of itsvelocity remains same and this is possible only whenthe particles moves in same straight line withoutreversing its direction.

(b) Non-uniform velocity : A particle is said tohave non-uniform velocity, if either of magnitude ordirection of velocity changes or both of them change.

(c) Average velocity : It is defined as the ratio of displacement to time taken by the body

taken TimentDisplaceme

velocityAverage = ;t

r v av

=

(d) Instantaneous velocity : Instantaneousvelocity is defined as rate of change of position vectorof particles with time at a certain instant of time.

Instantaneous velocityt

r v

t =

0lim

dt

r d =

(v) Comparison between instantaneous speedand instantaneous velocity

(a) instantaneous velocity is always tangential tothe path followed by the particle.

When a stone is thrown from point O then at pointof projection the instantaneous velocity of stone is 1v ,

at point A the instantaneous velocity of stone is 2v ,

similarly at point B and C are 3v and 4v respectively.

Direction of these velocities can be found out bydrawing a tangent on the trajectory at a given point.

(b) A particle may have constant instantaneousspeed but variable instantaneous velocity.

Example : When a particle is performing uniformcircular motion then for every instant of its circular

motion its speed remains constant but velocity changesat every instant.

(c) The magnitude of instantaneous velocity is equalto the instantaneous speed.

(d) If a particle is moving with constant velocity thenits average velocity and instantaneous velocity arealways equal.

(e) If displacement is given as a function of time,then time derivative of displacement will give velocity.

Let displacement 2210 t At A A x +=

Instantaneous velocity

)(2

210t At A A

dt

d

dt

x d v +==

t A Av 21 2+=

For the given value of t , we can find out theinstantaneous velocity.

e.g for 0=t ,Instantaneous velocity 1 Av = and

Instantaneous speed 1|| Av =

(vi) Comparison between average speed andaverage velocity

(a) Average speed is a scalar while averagevelocity is a vector both having same units ( m /s ) and

dimensions ][ 1LT .

2v

X O1v

Y

3v

CB

A

Fig. 2.6

4v


4/4

(b) Average speed or velocity depends on timeinterval over which it is defined.

(c) For a given time interval average velocity issingle valued while average speed can have manyvalues depending on path followed.

(d) If after motion body comes back to its initial

position then 0=av v (as 0= r ) but 0>av v and

finite as )0( > s .(e) For a moving body average speed can never be

negative or zero (unless )t while average velocity

can be i.e . 0>av v while av = or < 0.(f) As we know for a given time intervalDistance |displacement| Average speed |Average velocity|

Acceleration

The time rate of change of velocity of an object iscalled acceleration of the object.

(1) It is a vector quantity. Its direction is same asthat of change in velocity (Not of the velocity)

Table 2.2 : Possible ways of velocity change

When onlydirection of

velocitychanges

When onlymagnitude of

velocitychanges

When bothmagnitude and

direction of velocitychanges

Acceleration

perpendicular tovelocity

Acceleration

parallel or anti-parallel tovelocity

Acceleration has

two componentsone isperpendicular tovelocity andanother parallelor anti-parallel tovelocity

Ex. . Uniformcircular motion

Ex. . Motion undergravity

Ex. . Projectilemotion

(2) Dimension : [ M0L1T 2]

(3) Unit : metre/second 2 (S.I.); cm /second 2 (C.G.S.)

(4) Types of acceleration :(i) Uniform acceleration : A body is said to have

uniform acceleration if magnitude and direction of theacceleration remains constant during particle motion.

(ii) Non-uniform acceleration : A body is saidto have non-uniform acceleration, if either magnitude ordirection or both of them change during motion.

(iii) Average acceleration :t

v v

t

v a a

=

= 12

The direction of average acceleration vector is the

direction of the change in velocity vector ast

v a

=

(iv) Instantaneous acceleration =

dt

v d

t

v a

t =

=

0lim

(v) For a moving body there is no relation betweenthe direction of instantaneous velocity and direction of acceleration.

Ex. . (a) In uniform circular motion = 90 always

(b) In a projectile motion is variable for every point of trajectory.

(vi) If a force F acts on a particle of mass m , by

Newtons 2 nd law, accelerationm

F a =

(vii) By definition2

2

dt

x d

dt

v d a == =

dt

x d v As

i.e., if x is given as a function of time, second timederivative of displacement gives acceleration

(viii) If velocity is given as a function of position,then by chain rule

====dt

dx v

dx

d v

dt

dx

dx

dv

dt

dv a as.

(xi) Acceleration can be positive, zero or negative.Positive acceleration means velocity increasing withtime, zero acceleration means velocity is uniformconstant while negative acceleration (retardation)means velocity is decreasing with time.

(xii) For motion of a body under gravity,acceleration will be equal to g , where g is the

acceleration due to gravity. Its value is 2m/s8.9 or2cm/s980 or 2feet/s32 .

X O

Y

2

3

gg

g

1

a

a

a

Fig. 2.7

Documents

Motion in One Dimension Pt 1