MATHEMATI
CALPROOFOFAFEASI
BLEWARPDRI
VEANDTI
ME
TRAVELMECHANI
SM
I
NDEPENDENTRESEARCHER
DR.GAUTAM NARAYANBARUAH
ADDRESS:C/
OJ
.
N.BARUAH
J
ORHAT-785001
ASSAM,I
NDI
A
Phone:919678946027
Ema
i
l:ga
ut
a
mna
r
a
ya
nba
r
ua
h@gma
i
l
.
c
om
Abs
t
r
a
c
t:
Whe
nabodys
a
yat
oyc
a
rs
l
i
de
sdowna
ni
nc
l
i
ne
dpl
a
newi
t
ha
ngl
eof
i
nc
l
i
na
t
i
onl
e
s
st
ha
n45d
e
gr
e
et
hedownwa
r
da
ndf
or
wa
r
dmot
i
onoft
het
oyc
a
rc
a
nbe
r
e
s
t
r
i
c
t
e
db
yat
hr
e
a
dors
uppor
tpe
r
f
e
c
t
l
ypa
r
a
l
l
e
lt
ot
hehor
i
z
ont
a
l
.I
tc
a
nbema
t
he
ma
t
i
c
a
l
l
y
pr
ove
dba
s
e
dont
hel
a
wofc
ons
e
r
va
t
i
onofe
ne
r
gyt
ha
tapor
t
i
onoft
hewe
i
ghtoft
het
oyc
a
r
i
suns
uppor
t
e
de
ve
na
tt
hepos
i
t
i
onofr
e
s
ta
nds
i
nksdowni
nt
hes
a
mer
e
l
a
t
i
ves
pa
c
ec
a
us
i
ng
c
ompr
e
s
s
i
onofs
pa
c
e
t
i
mebe
l
owt
het
oyc
a
r.Thi
si
sama
t
he
ma
t
i
c
a
lpr
oofofapl
a
nes
o
l
i
d
s
ur
f
a
c
ewhe
r
ea
nobj
e
c
tc
a
nr
e
s
twi
t
hpa
r
tofi
t
’
swe
i
ghtnots
uppor
t
e
de
ve
na
tt
hepos
i
t
i
onof
r
e
s
te
na
bl
i
nganove
lwa
r
pdr
i
vea
ndt
i
met
r
a
ve
lme
c
ha
ni
s
mt
ha
tc
a
nwor
kwi
t
houtc
a
us
i
ng
a
nyphys
i
c
a
lha
r
mt
ohuma
ns
ubj
e
c
t
s
.
Ke
yWor
ds:
Wor
ki
ng wa
r
p dr
i
ve
;f
e
a
s
i
bl
et
i
met
r
a
ve
lf
orhuma
ns
ubj
e
c
t
s
;a
r
t
i
f
i
c
i
a
l
hype
r
s
pa
c
e
;t
i
mer
e
ve
r
s
a
l
.
Te
xt:
THEORYANDCALCULATI
ON
Ac
c
or
di
ngt
ot
hel
a
wofc
ons
e
r
va
t
i
onofe
ne
r
gy,e
ne
r
gyc
a
nne
i
t
he
rbec
r
e
a
t
e
d
norde
s
t
r
oye
d.Sowhe
nat
oyc
a
rCa
si
nf
i
gur
e1ofwe
i
ght‘
mg’
s
l
i
de
sdowna
ni
nc
l
i
ne
d
pl
a
nei
t
swe
i
g
hta
c
t
i
ngve
r
t
i
c
a
l
l
ydownwa
r
dsc
a
nbedi
vi
de
di
nt
oc
ompone
nt
s
(
a
)
(
mgc
os
θ+MKmgc
os
θ)pe
r
pe
ndi
c
ul
a
rt
ot
hepl
a
nes
ur
f
a
c
ea
c
t
i
nga
l
ongt
he
s
hor
t
e
s
tdi
s
t
a
nc
es
uppo
r
t
e
da
tr
e
l
a
t
i
ver
e
s
t
.
(
b)
(
mgs
i
nθ-MKmgc
os
θ)pa
r
a
l
l
e
lt
ot
hepl
a
nes
ur
f
a
c
ei
ni
ne
r
t
i
a
lmot
i
onwi
t
h
r
e
s
pe
c
tt
ot
her
e
s
tf
r
a
meofr
e
f
e
r
e
nc
e
.
Whe
r
eθi
st
h
ea
n
gl
eo
fi
nc
l
i
na
t
i
onoft
hepl
a
nea
nd‘
MK’i
st
hec
oe
f
f
i
c
i
e
n
tof
f
r
i
c
t
i
on.Fora
ni
nc
l
i
na
t
i
onoft
hepl
a
neo
fl
e
s
st
h
a
n4
5de
gr
e
et
hef
or
wa
r
da
nddownwa
r
dmot
i
on
oft
het
oyc
a
rc
a
nber
e
s
t
r
i
c
t
e
dbys
t
r
i
ngsT1 a
ndT2 pe
r
f
e
c
t
l
yp
a
r
a
l
l
e
lt
ot
hehor
i
z
ont
a
l
c
ons
i
de
r
i
nga
c
t
i
ona
n
dr
e
a
c
t
i
o
na
ss
i
mu
l
t
a
n
e
ousa
nde
xa
c
t
l
ye
q
ua
la
ndoppos
i
t
ei
na
c
c
or
d
a
n
c
e
wi
t
hne
wt
o
n'
st
hi
r
dl
a
wofmot
i
on
.
I
fMK i
st
hec
oe
f
f
i
c
i
e
ntoff
r
i
c
t
i
ont
he
nwe
i
ghtoft
hebodys
uppor
t
e
dby
f
r
i
c
t
i
oni
s(
MKxmgc
os
θ
)
.I
fMgEi
st
hewe
i
ghtoft
hebodynots
uppor
t
e
dbyf
r
i
c
t
i
ont
he
n
mgE=mg-(
MKxmgc
os
θ
)
.
Fi
gur
e2A -Le
tpl
a
neXY bei
nc
l
i
ne
dt
ot
hehor
i
z
ont
a
la
tθ.The
nl
i
neAB
pe
r
pe
ndi
c
ul
a
rt
opl
a
neXYi
si
nc
l
i
ne
dt
ot
hehor
i
z
ont
a
la
t(
90-θ)
.
Thewe
i
ghtoft
het
oyc
a
r
a
c
t
i
ngve
r
t
i
c
a
l
l
ydownwa
r
dsi
sr
e
s
t
r
i
c
t
e
dt
obet
r
a
ns
mi
t
t
e
da
l
ongl
i
neAB t
hes
hor
t
e
s
t
di
s
t
a
nc
et
ot
hei
nc
l
i
ne
dpl
a
nes
ur
f
a
c
e
mot
i
oni
sa
l
s
or
e
s
t
r
i
c
t
e
dbyahor
i
z
ont
a
lr
e
a
c
t
i
onf
or
c
e
.
.
Fr
omf
i
gur
e1a
nd2A
AB=mgc
os
θ
AX=mgc
os
θxs
i
n(
90θ)a
sf
or
c
e
sa
c
t
i
ngi
nt
womut
ua
l
l
y
pe
r
pe
ndi
c
ul
a
rdi
r
e
c
t
i
onsdonotc
a
nc
e
le
a
c
hot
he
r
.
Sot
hewe
i
g
htoft
het
oyc
a
runs
uppor
t
e
de
ve
na
tt
hepos
i
t
i
onofr
e
s
t
,
mguns
uppor
t
e
d=mg-[
(
MKxmgc
os
θ
)+mgc
os
θxs
i
n(
90θ)
]
0
I
ft
hewe
i
ghtoft
het
oyc
a
ri
s1kga
ndi
nc
l
i
na
t
i
onoft
hepl
a
ne30 t
ot
hehor
i
z
ont
a
l
t
he
nmguns
uppor
t
e
d=mg-[
MKxmgc
os
θ
)+0.
749956]
Somguns
uppor
t
e
di
spos
i
t
i
vei
fc
oe
f
f
i
c
i
e
ntoff
r
i
c
t
i
onMKi
sl
e
s
st
ha
n0.
25
.
Fr
om f
i
gur
e1a
nd2A pe
r
f
e
c
t
l
yhor
i
z
ont
a
lc
ompone
ntZY i
ss
uppor
t
e
da
nd
ve
r
t
i
c
a
lc
o
mp
one
ntZXi
suns
uppor
t
e
d,s
omguns
uppor
t
e
di
sma
xi
mumi
ft
hei
nc
l
i
na
t
i
onof
t
hepl
a
nei
s45de
gr
e
ea
si
nma
t
e
r
i
a
la
ndme
t
hods
e
c
t
i
on.
Duet
ot
hewe
i
ghtuns
uppor
t
e
dt
het
oyc
a
rmus
ts
i
nkt
owa
r
dst
hes
ol
i
ds
ur
f
a
c
e
i
nt
hes
a
mer
e
l
a
t
i
ves
pa
c
e
.Thi
smus
tt
e
ndt
oi
nc
r
e
a
s
et
hea
mountofe
ne
r
gype
runi
ts
pa
c
e
l
i
keac
ompr
e
s
s
e
ds
pr
i
ngpr
e
s
s
i
ngonas
ur
f
a
c
ewi
t
houta
nya
c
t
ua
la
ddi
t
i
onofe
ne
r
gy.Duet
o
t
hel
a
wofc
o
ns
e
r
va
t
i
onofe
ne
r
gyt
hea
c
t
ua
lwe
i
ghtf
a
l
l
i
ngve
r
t
i
c
a
l
l
yont
hei
nc
l
i
ne
dpl
a
ne
s
ur
f
a
c
ei
sa
l
wa
yse
qua
lt
omguns
uppor
t
e
d.Butduet
ot
hepr
e
s
s
i
nga
c
t
i
onl
i
keas
pr
i
ngt
he
r
e
a
c
t
i
onf
r
omt
hes
uppor
tmus
tt
a
kepl
a
c
eal
i
t
t
l
ee
a
r
l
i
e
rs
ot
ha
tt
hea
ppa
r
e
ntl
e
ngt
hoft
het
oy
c
a
rr
e
ma
i
nsuna
l
t
e
r
e
d.Thi
sme
a
nsove
rt
hes
a
mepe
r
i
odoft
i
met
hei
nc
l
i
ne
dpl
a
nes
uppor
t
e
d
apor
t
i
ono
ft
hewe
i
ghtoft
het
oyc
a
rove
ra
ni
nc
r
e
a
s
e
ddi
s
t
a
nc
e‘
D’
.
Sowor
kdoneore
ne
r
gyt
r
a
ns
f
e
r'
W'
=we
i
ghtuns
uppor
t
e
dxD-
Fr
om-Fi
gur
e1a
nd2A,
AB=(
We
i
g
htuns
uppor
t
e
dxD)c
os
θa
st
het
oyc
a
ri
snots
i
di
ngdowns
o
t
he
r
ei
snoc
oe
f
f
i
c
i
e
ntoff
r
i
c
t
i
on.
So,a
tt
hepos
i
t
i
onofba
l
a
nc
ea
ndr
e
s
tf
ort
het
oyc
a
r>
we
i
ghtuns
uppor
t
e
d
Si
n(
90θ)=
(
we
i
ghtuns
uppor
t
e
dxD)c
os
θ
we
i
ghtuns
uppor
t
e
d
we
i
ghtuns
uppor
t
e
dxD=
(
s
i
n90θ)x(
c
os
θ)
1
D=
(
s
i
n90θ)x(
c
os
θ)
Fr
omt
hee
qua
t
i
onf
orma
s
s
e
ne
r
gye
qui
va
l
e
nc
er
e
s
tma
s
s
>mo=
1xmguns
uppor
t
e
d
2
s
i
n(
90θ)xc
os
θxC
He
r
ewec
ons
i
de
rma
t
t
e
ra
se
ne
r
gya
l
r
e
a
dypr
e
s
e
nti
nt
her
e
s
tf
r
a
meofr
e
f
e
r
e
nc
ea
s
s
oc
i
a
t
e
d
wi
t
hgr
a
vi
t
y,
t
i
medi
l
a
t
i
on,
s
i
z
ec
o
nt
r
a
c
t
i
ona
n
da
c
t
i
ona
tadi
s
t
a
nc
e
.
Si
nc
ee
a
c
hpoi
nti
ns
pa
c
ei
sapa
r
t
i
c
ul
a
rpoi
nt
i
nt
i
me
-
The
nf
orma
t
t
e
rwi
t
hr
e
s
tma
s
st
he
r
ei
sma
t
he
ma
t
i
c
a
l
l
ymor
ea
mountofs
pa
c
e
t
i
mes
ot
ha
t
,
mguns
uppor
t
e
d
mo=s
pa
c
e
t
i
mex
2
C
Thi
si
nt
e
r
r
e
l
a
t
e
se
ne
r
g
y,ma
s
sa
nds
pa
c
e
t
i
me.
Fr
e
que
nc
ya
l
s
or
e
s
pons
i
bl
ef
o
r
pr
ope
r
t
i
e
sofma
t
t
e
ra
ndc
ons
c
i
ous
ne
sc
a
na
l
s
obede
f
i
ne
da
st
hea
mountof
t
i
mec
ont
a
i
ne
di
nawa
ve
.
Si
nc
ea
c
t
i
ona
ndr
e
a
c
t
i
ona
r
es
i
mul
t
a
ne
ousa
nd
e
xa
c
t
l
ye
qua
la
ndoppos
i
t
ei
na
c
c
or
da
n
c
ewi
t
ht
het
hi
r
dl
a
wofmot
i
ons
ot
he
s
pe
e
doft
i
mene
e
dst
obeac
ons
t
a
nti
na
l
lt
hei
ne
r
t
i
a
lf
r
a
meofr
e
f
e
r
e
nc
e
.
Si
nc
emome
nt
umofe
ne
r
gy=
E
C
Somome
nt
um MNwi
t
huni
tofKgme
t
r
epe
rs
e
c
ondge
ne
r
a
t
e
di
nt
hea
bove
c
ondi
t
i
on,
MN=
1xmguns
uppor
t
e
d
s
i
n(
90θ)xc
os
θxC
I
nf
i
gur
e2Bobj
e
c
tAi
si
nmot
i
onons
ol
i
dhor
i
z
ont
a
lpl
a
n
eXYwh
e
r
eapa
r
tof
i
t
’
swe
i
ghti
snots
u
ppor
t
e
de
ve
na
tt
hepo
s
i
t
i
onofr
e
s
t
.Le
tMVbet
hea
c
t
ua
lmome
nt
umoft
h
e
obj
e
c
ta
ndM i
t
’
sr
e
l
a
t
i
vema
s
s
.Wec
a
nnowc
a
l
c
ul
a
t
et
h
emome
nt
umMNoft
hepa
r
t
i
c
l
ef
r
omt
he
r
e
l
a
t
i
ons
hi
pa
bove
.
Si
nc
et
wof
or
c
e
si
nmut
ua
l
l
ype
r
pe
n
di
c
u
l
a
rd
i
r
e
c
t
i
ondono
tc
a
nc
e
le
a
c
hot
he
r
s
or
e
s
ul
t
a
ntmome
nt
umMRi
ns
o
l
i
dpl
a
neXYne
e
d
st
ober
e
ve
r
s
e
d>
2
2
2
(
MR)+(
MN)=(
MV)
MR=√{
(
MVxMV)
(
MNxMN)
}
Weknowt
ha
tamovi
ngbodyne
e
dspr
ogr
e
s
s
i
v
e
l
ymor
ef
or
c
et
os
pe
e
dup.
Ma
t
he
ma
t
i
c
a
l
l
ya
sa
bodys
pe
e
dsupt
hes
a
mef
or
c
et
e
ndst
oa
c
tbr
i
e
f
l
ywi
t
hde
c
r
e
a
s
eoff
r
e
que
nc
ypr
opor
t
i
ona
t
e
l
ype
runi
ts
pa
c
e
r
e
l
a
t
i
vet
oar
e
s
tobs
e
r
ve
r
.
He
r
et
hemot
i
ono
fe
ne
r
gyi
nt
hes
a
mer
e
l
a
t
i
ves
pa
c
ei
nc
r
e
a
s
e
st
her
e
l
a
t
i
vef
r
e
que
nc
y
ort
i
mec
ont
e
ntofaf
or
c
et
ha
tma
t
he
ma
t
i
c
a
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s
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s
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s
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a
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l
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a
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l
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mer
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tt
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e
nc
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.
Di
s
c
us
s
i
ona
ndc
onc
l
us
i
on:
Fr
omma
t
he
ma
t
i
c
a
lc
a
l
c
ul
a
t
i
onba
s
e
dont
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r
va
t
i
onofmome
nt
umwec
onc
l
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8
t
ha
twene
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dmor
et
ha
nCor3X10 numbe
rofknota
ndt
hr
e
a
duni
t
sa
sde
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r
i
be
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ma
t
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r
i
a
la
ndme
t
hods
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c
t
i
ont
oc
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t
r
uc
tawor
ki
ngwa
r
pdr
i
vea
ndf
e
a
s
i
bl
et
i
met
r
a
ve
lde
vi
c
e
f
orus
ebyhuma
ns
ubj
e
c
t
s
.
Thi
si
si
nf
a
c
ts
c
i
e
nc
et
oc
ons
i
de
rt
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i
t
i
ver
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s
ul
t
soft
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ma
t
e
r
i
a
la
ndme
t
hods
e
c
t
i
onwhi
c
hne
e
dst
obet
e
s
t
e
di
na
dva
nc
e
dl
a
bor
a
t
or
i
e
st
oahi
gh
de
gr
e
eofa
c
c
ur
a
c
y.
.
Re
f
e
r
e
nc
e
s:
1.
Ma
c
hE(
1872)Hi
s
t
or
ya
ndRootoft
hepr
i
nc
i
pl
e
soft
hec
ons
e
r
va
t
i
onofe
ne
r
gyOpe
n
Cour
tPub.Co.i
l
l
i
noi
s
.
2.
Va
nde
rWa
e
r
de
n,Bor
t
e
lLunde
r
t(
1983)-Ge
ome
t
r
ya
ndAl
ge
br
ai
nAnc
i
e
nt
c
i
vi
l
i
z
a
t
i
onSpr
i
nge
rI
SBN35
40121595
3.
Ei
ns
t
e
i
n,Al
be
r
t(
19050630)“
Ont
hee
l
e
c
t
r
odyna
mi
c
sofmovi
ngbodi
e
s
”Anna
l
e
n
de
rPhys
i
k17(
10):891921.
doi
:
10.
1002/
a
ndp.
19053221004.
4.
Ei
ns
t
e
i
n,Al
be
r
t(
1905)“Doe
st
hei
ne
r
t
i
aofabodyde
pe
nduponi
t
’se
ne
r
gyc
ont
e
nt
”
Anna
l
e
nde
rPhys
i
k18(
13):639641.doi
:
10.
1002/
a
ndp.
19053231314.
5.
Al
c
ubi
e
r
r
i
,Mi
gue
l(
1994)“
Th
ewa
r
pdr
i
ve:hype
r
f
a
s
tt
r
a
ve
lwi
t
hi
nge
ne
r
a
lr
e
l
a
t
i
vi
t
y”-
Cl
a
s
s
i
c
a
la
ndQua
n
t
umGr
a
vi
t
y11(
5):L73-L77.
doi
:
10.
1008/
0264_9381/
11/
5/
001
6.
Mor
r
i
s
,Mi
c
ha
e
l
;Thor
n
e
,Ki
p;Yu
r
ts
e
ve
r
,Ul
vi(
Se
pt
e
mbe
r1988)“Wor
mho
l
e
s
,Ti
me
ma
c
h
i
ne
sa
ndt
hewe
a
kEn
e
r
gyc
on
di
t
i
o
n”phys
i
c
a
lRe
vi
e
w61(
13):1
446-1449
7.
Vi
s
s
e
rM(
1995)Lor
e
nt
z
i
a
nWor
mhoe
s-Fr
omEi
ns
t
e
i
nt
oHa
wki
ngWoodbor
yNY:
AI
Ppr
e
s
s
8.
For
wa
r
dR.
L.(
1990)“
Ne
ga
t
i
vema
t
t
e
rpr
opul
s
i
on”J
our
na
lofpr
opul
s
i
ona
ndpowe
r
6:
28.
doi
:
10.
2514/
3.
23219.
9.
Ha
wki
ng,s
t
e
phe
n(
Apr
i
l27,2010)“
St
e
phe
nHa
wki
ng:Howt
obui
l
dat
i
me
ma
c
hi
ne
”Da
i
l
yma
i
l
10.
www.
a
nde
r
s
oni
ns
t
i
t
ut
e
.
c
omTi
meTr
a
ve
l
11.
I
ndi
a
npa
t
e
nta
ppl
i
c
a
t
i
on255/
KOL/
2015a
nd915/
kol
/
2015.
“
At
i
mewa
r
pa
ndf
r
e
ee
ne
r
gyma
c
hi
ne
”I
nve
nt
or-Ba
r
ua
hGa
ut
a
mNa
r
a
ya
n
12.
^
Pl
a
nk,
Ma
x(
1901)
"
Ont
hel
a
wofdi
s
t
r
i
but
i
onofe
ne
r
gyi
nt
henor
ma
ls
pe
c
t
r
um"
Anna
l
e
nde
rPhys
i
k4(
3)
:
553doi
:
10.
1002/
a
ndp19013090310.