UNI"ER*IDAD #O+ITECNICA DE+ GO+,O DE ME-ICO $Ciencia % tecnolo&ía 'ue trans(orman) DINAMICA
Estudiante: ANTONIO GARCIA HUERTA
Grupo 4° “B” EJERCIOS DE LOS TEMAS: Movimiento en un punto Movimiento en línea recta Movimiento curvilíneo Mecnica de or!itas Movimiento relativo
E"IDENCIA DE #RODUCTO
Paraíso, Tabasco !" #o$%"&br" !" '(4
2
−1
r/t01 / t 3 t580 i 3 /6 t
07
−2
./t01 v/t01/2t340 i56 t
7
PROB LEMA S TEMA 1
MO"IMIENTO EN UN #UNTO
PROB LEMA S TEMA 1
MO"IMIENTO EN UN #UNTO
296 La )ra*%ca !" +a $"+oc%!a! !" u- pu-.o "- *u-c%/- !"+ .%"&po "s u-a r"c.a0
Cua-!o .1 ' s, $1 4 p%"2s, 3 cua-!o .1 4 s, $1 ( p%"2s0 5a6 D"."r& D"."r&%-" %-" +a ac"+"r ac"+"rac% ac%//- !"+ pu-.o ca+cu+aca+cu+a-!o !o +a p"-!%" p"-!%"-." -." !" +a +í-"a r"c.a0 5b6 ob."-)a ob."-)a +a "cuac%/- para $ "- *u-c%/*u-c%/- !"+ .%"&po 3 7s"+a para para !"."r&%-ar !"."r&%-ar +a ac"+"rac%/- !"+ pu-.o0 DATOS: T1 ' s 1 4 *.2s T1 4 s 1 ( *.2s m=
S 2− S 1 2−T 1 T 2
m=
−10 −4 −14 =−7 ⟹ m= 4 −2 2
a =−7 a=
m s
2
dv dt
dv =−7 dt
∫ dv =∫ −7 dt v =−7 t + c v ( 2 )=−7 ( 2 ) + c 4 =−14 + c
Cua-!o .1 ' s, $1 4 p%"2s
18= c
v =−7 t + 18 a =−7
m s
2
29 +a pos%c%/- !" u- pu-.o !ura-." "+ %-."r$a+o !" .%"&po !" .1 ( s a .1 8 s "s S=
−1 2
3
2
t + 6 t + 4 t ( m )
0
a6 9Cu+ "s "+ !"sp+a;a&%"-.o !"+ pu-.o !ura-." "s." %-."r$a+o !" .%"&po< b6 9Cu+ "s +a $"+oc%!a! &=%&a !ura-." "s." %-."r$a+o !" .%"&po 3 "- >u? &o&"-.o ocurr"< c6 9Cu+ "s +a ac"+"rac%/- cua-!o +a $"+oc%!a! "s &=%&a< a6
Δ s =s 2− s1
s 1= 0
s 2= s 2=
−1 ( 2
−1 2
( 6 )3 + 6 ( 6 ) 2 + 4 ( 6 )
216 ) + 6 ( 36 ) + 24
s 2=−108 + 216 + 24 s 2=132
E-.o-c"s Δ s =132−0 1 @' D"sp+a;a&%"-.o !" .1( a .180
b6
S=
−1 2
3
2
t + 6 t + 4 t
ds = v dt v=
−3 2
2
t + 12 t + 4
v ´ =−3 t + 12
−3 t + 12=0
⟹
12=3 t
t =4 s
v = -3 (maximo) v= v=
−3 2
−3 2
3
t + 12 t + 4
( 4 )2 +12 ( 4 )+ 4
v =−24 + 48 + 4
m v max =28 s
c6
dv
'D.@ s1a = AB.C. dt
S35(6 1AB 5(6C 5(6'D 5(6@ a =− t + 12
8p%"s18C84D 546 44p%"s1 84C'D
a =− 3 ( 4 ) + 12 a =0 ( la velocidad esconstante) A1(
(4184C'8 2944 Supo-)a >u" s" r">u%"r" r"pr"s"-.ar +a pos%c%/- !" u- $"ícu+o >u" "s. 44184C'82 3 A + Bt + C t + Dt , !o-!" s%"-!o proba!o por &"!%o !" +a s"r%" !" po."-c%as S1 5.6 1 AB.C.'D.@ A, B, C 3 D so- co-s.a-."s0 E+ $"ícu+o par." !"+ r"poso "- .1( 3 S1(0 E- .14s, s1 8 *. 3 "- .1s, s 1 44 p%"s0 ' 1 B'C.@D. '81'8D a6 D"."r&%-" A, B, C 3 D0 1B'C 5(6 @D 5(6 ' 256 B6 9Cu+"s so- +a $"+oc%!a! 3 +a ac"+"rac%/- apro=%&a!as $"ícu+o "- .1s< D1 !"+ −256 B1( D1
Da.os: .1(
8p%"s1AB.C.'D.@ 1((546c 546' D 546 @
8p%"s18C84D 8p%"s18C8456
1((8C84D 8p%"s18C84 848p%"s18C 44p%"s1AB.C.'D.@ 1((56c 56' D 56 @ 1((84C'D
240 16
C1
1C
294; D"b" +a-;ar u- Fu")o !" ++a$"s a u- a&%)o >u" "s. "- "+ ba+c/- !" u-
s")u-!o pu-.o 0s% su"+.a +as ++a$"s a 0 & !"+ su"+o0 9A >u? $"+oc%!a! $"r.%ca+ s" -"c"s%.a para >u" ++")u"- a +a &a-o !" su a&%)o, >u" s" a++a a 8& sobr" "+ su"+o<
6m
m a =9.8 s
1.5 m
v =−9.8 ( 0.958 )
dv = a dt
v =−9.38 m/ s
Velocidad vertical resultado
dv = a (t ) dt
∫ dv =∫ a ( t ) dt v
t
v v0 =a ( t )¿
v −v 0= a ( t −t 0 )
v =v 0 +at ds = v 0 + at dt s
t
∫ ds =∫ ( v 0 + at ) dt s0
2
4.5 =4.9 t
0
2
s − s 0 =vot +
at
2
t =
2
4.5 4.9
t =√ 0.918
2
6 =1.5 + 0 +
9.8 t 2
t =0.918 s
2
6 −1.5=
9.8 t 2
292< +a $"+oc%!a! "- u- .r%-"o "s $1 ( . p%"2s s% "+ .1's su
pos%c%/- "s s1' p%", 9Cu+ "s +a pos%c%/- s% .1(s<
v =10 t pie / s
Tiempo en que llego de la mano al balcón.
v=
ds dt
ds =10 t dt
∫ ds =∫ 10 t dt 2
10 t s ( t )= 2
+c
s ( t )=5 t + c 2
25=20 + c
c =20 −25 c =−5
s ( t )=5 ( 10 ) + 5 2
s ( t )=500 + 5 s ( t )=500 pie / s
Pos%c%/- "- .1(0
292= u- au.o&/$%+ $%aFa a @( &%2 cua-!o s" "-c%"-!" +a +u; a&ar%++a !" u-
s"&*oro >u" s" "-cu"-.ra ' p%"s a!"+a-."0 La +u; a&ar%++a p"r&a-"c"r s a-."s !" >u" s" "-c%"-!a +a roFa0 a6 9Hu? ac"+"rac%/- co-s.a-." p"r&%.%r >u" "+ au.o&/$%+ a+ca-c" +a +u; "- "+ %-s.a-." "- >u" ca&b%" a +a roFa, 3 cu+ s"r +a $"+oc%!a! !"+ au.o&/$%+ cua-!o ++")u" a+ s"&*oro< b6 s% "+ co-!uc.or !"c%!" -o a+ca-;ar +a +u; a .%"&po 9Hu? ac"+"rac%/- co-s.a-." p"r&%.%r >u" "+ au.o&/$%+ s" !"."-)a Fus.o a-."s !" ++")ar a+ s"&*oro<
a6 a=
30
mi 5280 ft 1 h = 44 ft / s h 16 mi 3600 s
dv dt
v=
t
v
0
44
∫ a dt =∫ dv t 0
a t =v
v 44
a [ t −0 ] = v − 44
dv dt
ds =5 a + 44 dt 295
t
5
0
0
0
∫ ds=∫ 5 adt +∫ 44 dt 2
295= a
a t 2
( t −0 ) + 44 ( 5 −0 )
2
at = v −44
a t 295= + 220 2
v =at + 44
a=
295 −220 ( 2 ) 25
v =30 +44
a =6
v =74 ft / s
b6
( ) + 44
0 =a 5
a=
−44 5
a =−8.8 ft / s
2
296< cua-!o .1(, +a pos%c%/- !" u- pu-.o "s s18& 3 su $"+oc%!a! "s $1'&2s0 !"
.1( a .18s, su ac"+"rac%/- "s a1''. &2s !" .18s as.a >u" a+ca-c" "+ r"poso, su 2 ac"+"rac%/- "s a14 &2 s 0
a6 9Cu+ "s "+ .%"&po .o.a+ !" $%aF"< b6 9Hu? !%s.a-c%a .o.a+ cubr" "+ !"sp+a;a&%"-.o< 2
a =2 + 2 t a=
v
dv dt t
∫ dv =∫ (2 +2 t ) dt 2
2
0
2 3 v −2=2 t + t 3 2 3 v =2 t + t + 2 3 2
3
v ( 6 ) =2 ( 6 )+ ( 6 ) + 2 3
v ( 6 ) =158
v=
ds dt
2 t +
2 3
3
t
(¿+ 2 ) dt s
t
6
0
∫ ds =∫ ¿ 2 1 4 s =t + t + 2 t + 6 6
2 2 4 S ( 6 )=( 6 ) + ( 6 ) + 2 ( 6 )+ 6 3
S ( 6 )=270 m
a =−4 ⟹ a =
dv dt
v =−4 t + 182
−4 t =−182 t =
182 4
= 45.5 Segundos
s
t
270
6
∫ ds=∫ (−4 t + 182) dt
2
S =−2 t + 182 t + 72 −1092+ 270 2
S ( 45.5 )=−2 ( 45.5 ) + 182 ( 45.5 ) + 72−1092 + 270
S(45.5)= 3390.5 m
v =−4 t + 158 + 24 v
t
158
6
∫ dv =−4 ∫ dt v =−4 t + 158 + 24
PROB LEMA S TEMA 2
MO"IMIENTO EN UNA +INEA RECTA
296> La +a-ca !" +a *%)ura s" $a &o$%"-!o a '( ft / s , cua-!o su &o.or s" apa)a 2 !"b%!o a+a r"s%s."-c%a a"ro!%-&%ca su ac"+"rac%/- "s a =−0.1 v ft / s 0
9Cu+ "s +a $"+oc%!a! !" +a +a-ca Da.os: 2
a =−0.1 v ft / s 0
2s
!"spu"s<
t =20 ft / s
dv = a dt 2
v
0
20
2
v
∫−0.1 dt =∫ dv v 2
∫−0.1 dt =∫ v − dv 2
0
20
−0.2=
−0.2−
−1 v
1 20
1 −1 = 20 v
−0.25 =
v=
+
−1 v
−1 −0.25
v =−4 ft /¿
2982 La &a3or pro*u-!%!a! oc"-%ca !"scub%"r.a as.a aora s" a++a "- +as %s+as
Mar%a-as, "- "+ Oc?a-o Pac%*%co occ%!"-.a+0 -a bo+a !" ac"ro >u" s" +%b"r" "- +a sup"r*%c%" r">u%"r" 84 &%- para ++")ar a+ *o-!o0 La ac"+"rac%/- !" +a bo+a ac%a abaFo "s a =0.9 g −Cv , !o-!" ) "s +a ac"+"rac%/- !"b%!a a +a )ra$"!a! a+ -%$"+ −1 !"+ &ar 3 +a co-s.a-." C =3.02 seg 0 9Cu+ "s +a pro*u-!%!a! "- &%++as<
(
dv = a dt
ln 1 −
dv = 0.9−Cv dt v
∫ 0
e t
dv = dt 0.9 g −C 0
∫
ln (1−
1−
0.9 g
du =−Cdv
Cv 0.9 g
)=e−Cv
Cv =e−ct 0.9 g
−Cv u= 0.9 g −Cv
)
Cv =−Ct 0.9 g
−1 [
= e−ct −1
−Cv 0.9 g
=e−ct −1 ]
− −1 C
v
∫ 0
Cv =1 −e−Ct 0.9 g
t
du = dt u 0
∫
−1
( ln (u ) ) ] v =t ¿ t [ 0 0 C
0.9 g ( ln ( 0.9 g−Cv ) )− ln ¿=t −1 ¿ C 0.9 g
( ln ( 0.9 g−Cv ) )− ln ¿=−Ct
v=
0.9 g ( 1− e−ct ) C
v=
ds dt − ct
ds 0.9 g 0.9 g e = − dt C C u=−ct
¿
du =−cdt ln
(
0.9 g−Cv 0.9 g
)=−
Ct
s
t
0
0
∫ ds=∫
0.9 g 0.9 g −ct { − e } dt C C
0.9 g 0.9 g e [ s ] s =[ t + 2 0 C C
s=
0.9 g
C
−ct
0.9 g e
t +
2
C
(
m
0.9 9.81
s=
s
s
2
−1
3.02 seg
( =
−ct
8.829
m 2
s
)(
−1
3.02 seg
)(
[
[
−
64min = 3840seg g = 9.81m/s2 C = 3.02 seg -1
0.9 g 2
Evaluando t,C, g en !.
C
−(3.02 seg− )( 3840 ) 1
3840 seg ) +
S =11225.31 m
&% 1 8(0@44& 1 mi ] 1609.344 m
[
− ct
]
e 1 s= t + − C C C
] t 0
e
3840 seg )+ 0−
S =11225.31 m [
0.9 g
3.02 seg
−1
1 −1
3.02 seg
−
]
1 −1
3.02 seg
]
S =6.97 m 298 2
a =3 t m / s
2
- .r%-"o !" r".ro %&pu+s/ par." !"+ r"poso 3 ac"+"ra co-
as.a >u" su $"+oc%!a! "s !"
1000 m / s
a =−0.002 v
"-cu"-.ra u- *r"-o !" a)ua 3 su ac"+"rac%/- "s !" $"+oc%!a! !%s&%-u3" a
500 m / s
"- "s" &o&"-.o 2
as.a >u" su
0
9Hu? !%s.a-c%a .o.a+ r"corr" "+ .r%-"o<
v =0 m / s
v =1000 m/ s
dv = a dt
dv = 3 t 2 dt
,
1000
t
2
dv =3 t dt
, 3
∫ dv =∫ 3 t dt 2
0
v =500 m / s
t 1000=3 3
,
0
3
1000=t
,
Ca+cu+a&os .%"&po 3
t =1000
t =√ 1000 3
,
t =10 s
,
Por +o .a-.o v
t
3
∫ dv =∫ 3 t dt 2
0
,
0
t v =3 3
3
v =t
,
Ca+cu+a&os pos%c%/- !"+ pr%&"r .ra&o s
t
∫ ds=∫ t dt 3
0
0
4
,
s=
,
v.
t
4
4
10 s 10= 4
,
,
s =2500
'!o .ra&o dv ds . =a dt ds
dv =a ds
,
v.
dv =−0.001 v 2 , ds
v.
dv v
=−0.001 ds
2
s
500
dv v . 2 = −0.001 ds v 2500 1000
∫
ln
∫
500 1000
=−0.001 s + 2.5
ln 500− ln1000 =−0.001 s + 2.5
,
,
1 ln − 2.5=−0.001 s 2
ln
,
1 2
−2.5
−0.001
=s
s =3193.1
298? E+ a-+%s%s !" &o$%&%"-.o !" u- &"ca-%s&o %-!%ca >u" +a $"+oc%!a! !" u-
pu-.o !" co-"=%/- "s. !a!a por
2
v = A + 4 s ft / s , !o-!" A "s u-a co-s.a-."0
2 Cua-!o s =2 ft , su ac"+"rac%/- "s a =320 ft / s . 9Cu+ "s su $"+oc%!a! cua-!o
s =2 ft
a=
<
dv ds dv ds dv . = . =v dt ds ds dt ds
v = A + 4 s
2
dv =8 s ds a ( s )= ( A + 4 s ) ( 8 s ) 2
2
A + 4 ( 2 )( 8 ( 2 )) a ( 2 )=¿ a ( 2 )=( A+ 16 ) (16 ) 320=16 A + 256
320−256 =16 A 64 = 16 A
A = 64 / 16
A = 4 2
v ( s )= 4 + 4 s
2
v ( 2 )= 4 + 4 ( 2) v ( 2 )= 4 + 16 v ( 2 )=20 ft / s
29=4 La ac"+"rac%/- !" u- cu"rpo "s. !a!a por ft
a =3 s
ft
2
s
2
$"+oc%!a! "s v =10 s 0 9Cu+ "s su $"+oc%!a! cua-!o s = 4 ft <
dv 2 v =3 s ds
v
s
∫ vdv=∫ 3 s ds 2
10
0
v
2 3
' 2 − 50= s K
0 E-
s =0, su
v
2
1
2s
3
100
v =√ 2 s + 100 3
v ( 4 )=√ 2 ( 4 ) + 100 3
v ( 4 )=√ 2 ( 64 ) + 100 v ( 4 )=√ 128 + 100 v ( 4 )=√ 228 v ( 4 )=15.09
ft s
29=8 E- "+ prob+"&a '0@ supo-)a >u" "-
t =0
+a &asa s" +%b"ra !"s!" "+
r"poso "- +a pos%c%/- s =1 m 0 D"."r&%-" +a $"+oc%!a! !" +a &asa "- *u-c%/- !" S
a+ &o$"rs" !" su pos%c%/- %-%c%a+ a s =0 0
s =0
a =−45
s =0 m
v =1
m s
v =0
s =1 m
v
s
0
1
∫ vdv=−4 ∫ sds 2
v =−2 s2 + 2 ( 1 )2 2 2
v =−2 s2 + 2 2 v =± √ 4 − 4 s
2
v =± 2 √ 1− s
2
v =−2 √ 1− s
m s
2
m s
m s
29> La Ac"+"rac%/- !" u- P+a-".a %po.?.%co B%!%&"-s%o-a+ !"p"-!"ría !" +a
!%s.a-c%a s !"s!" "+ c"-.ro !"+ p+a-".a s")7- +a r"+ac%/-
a=
−k s
, !o-!" "s
u-a Co-s.a-.", s"a R" "+ ra!%o !"+ p+a-".a 3 G" +a &a)-%.u! !" +a ac"+"rac%/!"b%!a a +a )ra$"!a! "- su sup"r*%c%" 5a6 S% u- cu"rpo s" +a !a u-a o ac%a a*u"ra a u-a !%s.a-c%a So !"s!" "+ c"-.ro !"+ p+a-".a, !"."r&%-" su $"+oc%!a! "- *u-c%/- !" S0 −k a= s
( )
dv ds −k = dt ds s dv −k v = ds s
dv v =
−k s
ds
V
S
Vo
So
∫ v dv =∫ −sk ds v
2
2
−
2
vo 2
=−k ( ln ( S )− ln ( So ))
v =√ vo −2 k ( ln ( S )− ln ( So ) ) 2
v =√ vo −t!t ( ln ( S )− ln ( So ) ) 2
PROB LEMA S TEMA 3
MO"IMIENTO CUR"I+INEO
296 Los co&po-"-."s !" +a ac"+"rac%/- !" u- pu-.o "- *.2s ' so- a=1 @.', a 31 8. 3
a;1 (0 E- .1 (, =1 *., =1 @ *.2s, 31 *., 31 ' *.2s, ;1 ( 3 ;1 (0 9Cu+"s so- sus $"c.or"s !" pos%c%/- 3 !" $"+oc%!a! "- .1 @ s<
a@1 6t2 dVx ' dt 1 @. Vx
t
∫ dVx
1@
3
∫ t 2 dt 0
dSx @ dt 1 5. @6 Sx
∫ dSx 5
t
1
" # -3= t 3 ! # -5= 3
V x = t + 3 S x =
∫ (t 3 +3 ) dt 0
t 4 4 + 3t
t 4 4 + 3t + 5
"@1 t6 5 6
dV# dt 1 (
a%1 t dV" dt 1 8.
V#
∫ dV# V"
∫ dV" −2
t
18
0
t
1(
∫ dt 0
∫ t dt 0
V z= 0
2 " + 2=3 t
V y =3 t 2 -2
"%16 t2 3 2 dS" ' dt 1 5@. '6
a1 <
S"
∫ dS" 1
t
"1 <
∫ (3 t 2−2 ) dt
1
0
dS# dt 1 (
S31 .@'. S#
Sy= t3-2t+1
∫ dS# 0
t
1
∫ 0 dt 0
Sz= 0
=5@61 5@6@@1 @( 35@61 @5@6''1 ' ;5@61 ( $
i^ 5 2= %^ 5 <
V 1 6<
S=5@61
(3 )
⁴
4
@5@61
^ k
137 4
S35@61 5@6@'5@61 '' S;5@61 ( $
S 1
137 4
i^ 5 22 %^ 5 <
^ k
29;< u- ba."a!or )o+p"a +a p"+o.a a @ *. sobr" "+ coFí- !" o&" 3 +a "+"$a co- u-
-)u+o !" 8(N sobr" +a or%;o-.a+0 E+ '!a bas" +a .o&a a 8 *. sobr" +a '!a bas"0 9Cu+ *u" +a $"+oc%!a! %-%c%a+ !" +a p"+o.a<
v x =v 0 x a x =0 dv x dt v x
∫ v 0 x
ds x dt
= v 0 cos &
=0 & v 0 cos ¿ dt
t
∫
d vx = 0
0
dt
¿ ¿
t
∫¿
ds x =
0
v x −v 0 x =0
s x
∫¿ 0
s x =v 0 cos & t
2
s " −3 =v 0 sin & t −
g t 2
2
2
2
a + ' =c 2
90
gt s " = v 0 sin & t − +3 2
2
+ 902 =c 2 127.27 = v 0 cos & t
c =√ 16200 c =127.27 ft
t =
127.27 254.54 = v 0 cos & v0
v 0 t =254.54
a " =−g
2
dv " dt
gt 6 =v 0 sin & t − +3 2
=−g
2
v "
∫d
t
v"
=−g ∫ dt
v 0 "
gt 0 =v 0 sin &t − −3 2
0
2
v " =v 0 " −¿ v (¿¿ 0cos & −¿) ds " =¿ dt
gt 0 =254.54 sin 60− −3 2
− g t 2 2
2
t =
=3 −220.43
217.43 16.1
v 0 sin &−¿¿ dt
¿ ¿
t =√ 13.5 t
∫
ds " = ¿ 0
t =3.67
s x"
∫¿ 3
127.27= v 0 cos60 ( 3.67 )
v0 =
v0 =
127.27 cos 60 ( 3.67 ) 127.27 1.835
v 0 =69.35 ft / s
29;8 - ;o/+o)o "s. pro$%s.o !" u- arco 3 u-a *+"ca >u" .%"-" u-a F"r%-)a co-
.ra->u%+%;a!or, pu"s !"b" &"!%r +a ."&p"ra.ura !" u- r%-oc"ro-."0 E+ a+ca-c" &=%&o "s !" 100 m0 S% "+ r%-oc"ro-." "&b%s." !%r"c.a&"-." ac%a "+ ;o/+o)o a 30 km/h 3 ?s." apu-.a su arco 20 sobr" +a or%;o-.a+, 9A >u? !%s.a-c%a !"b" "s.ar "+ r%-oc"ro-." cua-!o !%spar" +a *+"ca<
(
30 km h
)(
1000 m 1 km
Vox =VoCos& Vo" =VoSen& d( x dt
)(
1h 3600 s
)=
8.33 m / s
d( " dt
=v "
d( " = v 0 "−¿ ( "
=V x
t
∫ d( =∫ v "
0
dt
0
0 "
−¿ dt
d( x
2
VoCos&dt
=
x
t ( " = v 0 cos ( 20 )− g 2
t
∫ d( =∫ VoCos (20 ) dt x=
31.304 sen (20 ) =1.09 s 9.8
0
t hmax=
VoCos ( 20 ) t
t 1 =2.19 s
x
0
V x = 8.33 m / s d( x =¿
Retomando :
a x dt
( x =8.33 ( 2.18 )=18.16
lcance max. !n "5# 100= v o cos ( 45 ) t
t
d( x =¿
∫ 8.33 dt 0
0 =v o sen ( 45 ) t −
1 ( 9.8 ) t 2 2
0 =v o sen ( 45 )
100
x
∫¿ 0
VoCos ( 45 )
( x =8.33 t 2
V 100 tan ( 45 )=
a " =−g dv " =−gdt V "
2
v=
(
100
VoCos ( 4
4.9 ( 10000)
( cos (45 ))2
9800 100
t
∫ dv =∫− gdt
v
"
v 0 "
V "
1 2
− ( 9.8 )
2
=$%&
0
V 0=31.30 =
v 0 "
- gt
m's
x =31.304 cos (20 ) ().1%*
x =64.12 m+ lo )ue((eco((e el (ine(onte +
¿ 82.28 m
29;= +os c+a$a!%s.as !" +a >u"bra!a "- Acapu+co !"b"- s%-cro-%;ar !" &o!o >u"
"-.r"- a+ a)ua "- +a cr"s.a !" u-a o+a0 Las cr"s.as !" +a o+as .%"-"- 'p%"s sobr" +a pro*u-!%!a! &"!%a 1' p%"s !"+ a)ua +a $"+oc%!a! !" +as o+as "s
√ gh 0 La &".a
!" +os c+a$a!%s.as "s u- pu-.o a 8 p%"s !" +a bas" !"+ aca-.%+a!o0 Supo-)a >u" cua-!o s" %-%c%a "+ c+a$a!o +a $"+oc%!a! "s or%;o-.a+0 5a6 9Cu+ "s +a &a)-%.u! !" +a $"+oc%!a! "- &%2 cua-!o "-.ra- a+ a)ua< 5b69A >u? !%s.a-c%a !" +a &".a !"b" "s.ar +a cr"s.a !" +a o+a cua-!o s" +a-;a uc+a$a!%s.a para >u" "-.r" a+ a)ua sobr" "++a<
,lavadista a x
a "=− g
=&
ds x
= v 0 cos&
dt
dv x dt
dv "
=0
dt
5 x
=−g
t
∫ ds =( v cos& )∫ dt 0
x
0
0
v x
t
v x 0
0
∫ ¿ 0∫ dt 5 x =v 0 cos&t
v "
∫ dv =− ∫ t
v 0 "
"
g
o
dt
v x =v
v " = v
0 " −¿
x 0
ds " dt
=−¿ " =¿ v 0 sen& −¿ v¿
s "
t
∫ ds =−g ∫ tdt "
s0 "
0
v " =−¿ s " =
−g 2
2
t + s 0 "
v0 =
5 x
)=-
cos&t
2
16.1 t %.5 27 v0 = 2.3 2
t =
−85.5 −16.1 v 0 =11.7
t=
√ 5.31 v x =v 0 cos0 v x =v 0 v x =11.7
t=)./
) 0t !l impacto ocurre en impacto=".$
s " =2
ft s
Tiempo de impacto=)./ v " =−32.2 t
Velocidad de
1mi=5)%&0t ft ".$ s
1 mi 5280 ft
3600 s mi =51.1) h 1h
v " =−74.06
√ v x 2
V=
ft s 2
v "
ft V=".$ s
olas
√ gh =√ 32.2=19.65 ds x
= 19.65
dt 5 x
t
∫ 5 =19.65∫ dt x
0
5 x =19.65 t
Tiempo de impacto=)./ 5 x =45.19 ft
a cresta se debe encontrar a "5.1$0t
0
ft s
29;> u-a bo+a !" c"ro "- u- .a->u" !" ac"%." s" +" !a, "- .1(, u-a $"+oc%!a! m
or%;o-.a+ $1'% &2s, +as co&po-"-."s !" su ac"+"rac%/- "v x
, a " 1 0' v " ,
a #
s
a x
so-
2
1 0' v # 0 9Cu+ "s +a $"+oc%!a! !" +a bo+a "-
.1s<
v =2 i ⃗
.1(
v "
∫
8 v " + 1.2
0
dt v x
∫ 2
(¿ ¿ " + t
d v x =−1.2 dt dt 0
∫
-5 ln (
e
v x ) 2
6I-5'61 0'.
=e
−1.2 t
v " =
a " =−8−1.2 v "
+ 1= e−1.2 t
d v # dt
5
8 −1.2t ( e −1 ) 1.2
a #=−1.2 v #
=−8 −1.2 v " 1 −1.2
= e−1.2t
1.2 = e−1.2t −1 8
v x =2 e
dt
( )=−
( )
8
= e−1.2t
d v "
e
1.2 v "
−1.2t
v x 2
0
8 8 )−ln 1.2 1.2 ln ¿
8 1.2 8 1.2
v " +
ln
l
=−1.2∫ dt
v
=−1.2 v x
v x
t
d v "
a x =−1.2 v x d v x
d v " dt
¿=
8 1.2
+ v "
10'
∫
=−1.2 v #
d v # dt
=−1.2∫ dt
1.2 t
e
ln ( v # )
= e−1.2 +c −1.2t
$
v #= c e v #
v =0.602 ˇi − 4.65 %ˇ
1.2 (0 )
(0 )
=c e−
(1c −1.2 t
v #=0 ( e
)
v #=0
v #=0
t=&
Velocidad en t=1 −1.2 ( 1)
v x =2 e
v x =0.602
v " ( )= 1
v " =−4.65
8 −1.2 ( 1) (e −1 ) 1.2
2
29>4 S%
d" mm d " " =150 mm * = 300 *" =0 2 dt s d t
9Cu+"s so- +as &a)-%.u!"s !" +a $"+oc%!a! 3 +a ac"+"rac%/- !"+ Pu-.o P<
2
2
2
2
2
2
! = x + "
x = ! − "
x =√ ! − " 2
2
x =√ 0.3 −0.15 2
2
dx mm =−0.1732 dt s
x =0.2598 mm
2
2
2
d! dx d" = + dt dt dt
0 =2 x
2
dx d" + 2 " dt dt
dx d" 0 =2 x + 2 " dt dt
2
d x =−0.3464− 0.1154−0 2 d t 2
( )
2 dx =− " d" 2 x dt dt
()
dx " d" =− dt x dt
(
( ) ( )( ) ( ) 2
2
d x −1 d" " d" dx " d " = + 2 − 2 x dt x dt dt x d t 2 d t
d x mm =−0.4618 2 2 d t s
|V |=√ ( 0.3 )2+ (−0.1732 )2
)
0.15 dx =− ( 0.3 ) 0.2598 dt
|V |=0.3464 mm
|a|=0.4618 mm 2
s
s
|a|=√ ( 0 )2 + (−0.4618 )2
29>2 - au.o&/$%+ $%aFa a ((&2 sobr" u- ca&%-o r"c.o co- p"-!%"-." cr"c%"-."
cu3o p"r*%+ $"r.%ca+ s" pu"!" apro=%&ar co- +a "cuac%/- &os.ra!a0 Cua-!o +a coor!"-a!a or%;o-.a+ !"+ au.o&/$%+ "s =14((&, 9Cu+ "s su ac"+"rac%/-< 100 km / h =).% m's
c =0.0003 " = c x
√( ) ( ) 2
v=
dx d" + df df
2
( )
d" d" dx = dt d" dx d" d" dx = dt dx dt
( )
d" dx = 2 cx dt dt
√ ( ) ( ( )) 2
v=
dx dx + 2 cx df dt
2
2
v=
√( )
v=
dx 2 1 + ( 2 cx ) √ dt
2
dx + ( 1+ 2 cx )2 df
dx v = dt √ 1 + ( 2 cx )2 27.78 dx = dt √ 1 + ( 2 x )2 x 2
vx ( 400 ) =
27.75
√ 1 + 0.0376
=27.01
dvx dvx dx = dt dt dx
1 + ( 2 cx )
2
¿
[(
dvx −1 ( ¿ −3 ) 2 =v + ¿ 8 c x 2 dt 2 1 + ( 2 cx )
2
¿ 2 dvx −4 c vx = vx dt −3 ( ¿¿ ) 2
) ]( ) dx dt
2 ( 0.0003 ) 400
¿ ¿ ¿2 1 +(¿ ¿)¿ √ ¿ ¿ ax ( 400 )=− 4 ( 0.0003 ) ( 100 ) ( 400 ) ¿ ¿ 2
ax ( 400 )=−0.0993 m / s 2
2
( )( ) ( ) 2
d " d x dx dx = 2 cx 2 + +2c 2 dt dt dt dt 2
d " = 2 ( 0.0003 ) ( 400 ) (−0.0993 )+( 27.01 )2 + 2 ( 0.0003 ) 2 dt 2
d " =−0.023832+ 0.43772406 2 dt a" ( 400 ) =0.41389206 a =−0.0993 i^ + 0.4138 %^ ⃗
PROB LEMA S TEMA 4
MECANICA MECANICA DE ORBITA* ORBITA*
.%"rra a u- pu-.o *%Fo sobr" "+ "cua!or, 3 29> E- +a *%)0 '08, s"a L u-a +í-"a !"+ c"-.ro !" +a .%"rra ,0 s"a
u-a +í-"a !" r"*"r"-c%a !" !%r"cc%/- *%Fa0 La *%)ura &u"s.ra a +a T%"rra $%s.a !"s!"
arr%ba !"+ po+o -or."0 9Es
d& dt pos%.%$a o -")a.%$a<
9Cu+ "s +a &a)-%.u! !"
d& dt "-
(ad s <
So+uc%/-: A6
B6
d& >0 dt
2 -(ad d& −5 (ad = 10 dt 24 ( 3600 ) s 1 0'= s
29>> 29>> E- +a *%)ura P'0, "+ -)u+o Q "-.r" +a barra 3 +a +í-"a or%;o-.a+ "s Q 1 3
2
t −2 t 4 5)ra!os60 D"."r&%-" +a $"+oc%!a! 3 +a ac"+"rac%/- a-)u+ar"s !" +a
barra "- . 1 ( s0
3 2 Q 1 t −2 t 4
1
d& dt
2 1 3 t 4 t
( 10)
1
4 (10 )
2
3 ( 10 )
"+oc%!a! "+oc%!a! A-)u+ar ( 10)
1
1 '8( &2s d dt
1
8 t
4
Ac"+"rac%/- A-)u+ar / ( 10)
1 8 ( 10 ) 4 / ( 10) 1 8 &2 s
2
29>? La ac"+"rac%/- a-)u+ar !" u-a +í-"a $ r"sp"c.o a u-a +í-"a !" r"*"r"-c%a 2 ,0 "s 1 @( 8 t ra!2 s 0 Cua-!o t 1 (, Q 1 ( 3 1 (0 9Cua+ "s +a
$"+oc%!a! &=%&a !" $ r"sp"c.o a , !ura-." "+ %-."r$a+o !" .%"&po !" t 1 o 0
a t 1 ( s") d dt
1
∫d 0
dt 1 d t
1
∫ 30 0 6 t dt 0
1 @( t −3 t 2
D"r%$a&os para "-co-.rar -u"s.ros pu-.os crí.%cos 1 1 1
1 @( 8 t 1 8
Sus.%.u%&os t 1 "- para ob."-"r -u"s.ra $"+oc%!a! &=%&a0
2 1 @( t −3 t
1 @( ( 5 )−3 ( 5 )2 =75 (ad / s
"+0 M=%&a
29?< -a .urb%-a !" )as "&p%";a a )%rar "-
t =0 co- ac"+"rac%/- a-)u+ar 1 8
t ra!2 s 2 !ura-." @ s 3 +u")o !"sac"+"ra co- 1 @ ra!2
s
2
!".%"-"0 a6 9Hu? $"+oc%!a! a-)u+ar &=%&a a+ca-;a< •
•
•
1 8 t ra!2 s . 1 ( @ s0 d dt
1
dt 1 d
•
•
•
•
•
•
2
t
∫d 0
∫ 6 t dt
1
0
2 1 3 t
( max )
1 ( f ) ( i)
( max )
1 ' (
( max )
1
( 0)
1
27
ra!2s 2
3 (0 )
1 (
( 3)
1
2
3 (3 )
1 '
as.a >u" s"
29?2 La a)uFa !" u- %-s.ru&"-.o !" &"!%c%/- "s. co-"c.a!a a u- r"sor." .ors%o-a+ 2 >u" +a so&"." a u-a ac"+"rac%/- a-)u+ar 1 4 Q ra!2 s , !o-!" Q "s +a
pos%c%/- a-)u+ar !" +a a)uFa "- ra!%a-"s r"sp"c.o a u-a !%r"cc%/- !" r"*"r"-c%a0 S% +a a)uFa s" +%b"ra !"+ r"poso "- Q 1 ra!, 9Cu+ "s su $"+oc%!a! a-)u+ar "- Q 1 (< •
d dt 0
d& d & 1
•
•
•
•
•
•
&
∫ d
1
0
∫ −4 & d & 1
1
d& dt 1 1 2
1 2 1 2
2
2
2
1 ' ( 1)
1 '
1 5'6 5
1 2
614
√ 2 1 √ 4
1 ' ra!2s
d d & 0
2
Para &=1
2
•
d d & 0
"+oc%!a! A-)u+ar
2 1 ' &
1 4 Q
29?6 E+ -)u+o Q " +a *%)0 '0@ &%!" +a !%r"cc%/- !"+ $"c.or u-%.ar%o ! r"sp"c.o a+ "F" =0 S% 1 !Q2! t 1 ' (ad / s , !"."r&%-" "+
$"c.or !!2! t : a6 Cua-!o Q 1 ( b6 cua-!o Q 1 (° c6 cua-!o Q 1 (°0 •
•
1
d& dt 1 '
d2 dt 1
d& dt U 1 U
a% •
•
&
1 (°
d2 dt 1 ' V
&% •
&
1 (°
(ad / s
d2 dt 1 ' W
•
c6 & 1 (°
•
d2 dt 1 ' V
•
29>? La ac"+"rac%/- a-)u+ar !" u-a +í-"a , r"sp"c.o a u-a +í-"a !" r"*"r"-c%a ,0
"s
Cua-!o r"sp"c.o a
(ad
=30− 6 t
∝
t =0, &=0
,0
d3 = ∝ dt d3 = 30 −6 t dt
2
s
3
.
3 =0.
9Cu+ "s +a $"+oc%!a! a-)u+ar &=%&a !" ,
!ura-." "+ %-."r$a+o !" .%"&po !" t =0 a t =10 s 4
t
∫
d3 =¿ ( 30 −6 t ) dt 0
3
∫¿ 0
2
3 =30 t −3 t 1
3 =30−6 t 30−6 t = 0
6 t =30
t =
30 6
t =5 1 1
3 =−6 3
1 1
(5 )=−6
3 ( 5 ) =30 (5 )−3 ( 5 )
2
3 =75 ms velocidad m@ima
PROB LEMA S TEMA 5
MO"IMIENTO RE+ATI"O
294<=9 La &a)-%.u! !" +a $"+oc%!a! !"+ a$%/- &os.ra!o "s co-s.a-." " %)ua+ a 400
m s 0 La ra;/- !" ca&b%o !"+ -)u+o Q !" su .ra3"c.or%a "s co-s.a-." " %)ua+
a °2s0 a6 9Cu+"s so- +a $"+oc%!a! 3 +a ac"+"rac%/- !"+ a$%/- "- .?r&%-os !" co&po-"-."s -or&a+ 3 .a-)"-c%a+<
b6 9Cu+ "s "+ ra!%o !" cur$a.ura %-s.a-.-"o !" +a .ra3"c.or%a !"+ a$%/-<
| |
5 5 - s 180 5
1
d& = 5 5 / s dt
v ⃗
v ⃗
S 1 XQ
-
ds =¿ dt
36
1 $ e^ 1 4((
.
$1 X Y
e^
6=
v 3
6=
400 - 36
.
d ⃗v d ^e t ( 400 ) = dt dt
a^
a^
a^
1 (
1(
- 36
d e^ t = 3 ^e dt
m 400 s
3=
=
e^ . Y $ e^ -
e^
.
( )(
- 400 ) ^e 36
1 ( e^ . @40 e^ .-
X
d& dt
(ad X 1 4@088'@8 s
294<9 E- . 1(, u- au.o&/$%+ par." !"+ r"poso "- "+ pu-.o A, s" &u"$" ac%a +a m
!"r"ca 3 +a co&po-"-." .a-)"-c%a+ !" su ac"+"rac%/- "s a. 1 (04 s 2 0 9Cu+ "s +a &a)-%.u! !" +a ac"+"rac%/- !"+ au.o&/$%+ cua-!o ++")a a+ pu-.o B<
dv
a. ¿ dt
v
∫ 0.2 t
2
t
∫ tdt
1(04
0
0
2 1 (0' t
ds =0.2 t 2 dt s
∫ ds=0.2 ∫ t dt 2
S 1 XQ
0
- S =50 2
⃗ a
3
. 1 80(8s
d& 2 0.2 t e^ e^ . . dt
1 (04. e^ .
0.2 3 t 3
t =4178.097245 e^ .
2 1 (0' t
0.2 3 t 3
'0@1
SB 1'0@&
d ⃗v = 0.4 t dt
0
S=
SB 1 ( 200 + 25 - ) m
v ⃗
t
( 0.2 t 2 )2 ^ 50
e -
S 1 XQ ds d& = 6 dt dt
⃗ a ( 16.10619111 )=64.44247644 ^e . @0@48 e^ -
d& v = dt 6
|a|=54.21
u- pro3"c.%+ .%"-" u-a $"+oc%!a! %-%c%a+ !" '( p%"s2s a @(° sobr" +a or%;o-.a+0 2944<
a6 9Cu+ "s +a $"+oc%!a! 3 +a ac"+"rac%/- !"+ pro3"c.%+ "- .?r&%-os !" +as co&po-"-."s -or&a+ 3 .a-)"-c%a+ cua-!o "s. "- "+ pu-.o &s a+.o !" su .ra3"c.or%a< b6 9Cu+ "s "+ ra!%o !" cur$a.ura %-s.a-.-"o !" +a .ra3"c.or%a !"+ pro3"c.%+ cua-!o "s." s" "-cu"-.ra "- "+ pu-.o &s a+.o !" su .ra3"c.or%a< Op"rac%/-: a6 1 '( *.2s
V "
∫ Vo "
Q1@(°
a x =0
Vo x =20cos30
Vo " =20cos30
a "=−9.81
Vo x =17.32
Vo " =10
d V " dt 1
t
∫−9.81 0
V " =−9.81 + V "
(0.
a6 1'( cos@(".1 0@' ". b6 a10 "a
V 2 7 = 8
2
V 8= an
1
(17.32)2 9.81
=30.7792
294<<9 S" .%"-" "+ -)u+o Q1 '
2
t ra!0
a6 9Cu+ "s +a &a)-%.u! !" +a $"+oc%!a! 3 !" +a ac"+"rac%/- !" P r"sp"c.o a O "- .1 s< b6 9Hu? !%s.a-c%a a +o +ar)o !" +a .ra3"c.or%a c%rcu+ar r"corr" P "-.r" .1( 3 .1 s< So+uc%/-: 2 Q1 ' t
d& = 4 t =3 dt 2 (ad d & = / 2 2 d t 14 s
2 S1r Q14 Q1 t
v t =16t m /s
1r Y1454.61 8 . at
dv 2 16 m / s = 1 dt
a6 $1856". &2s1 8 ". 5&2s6 2
a1 !/et + ! 3 "# 2
2 a1546546". 5465 4 6"# 5&2 s 6
a18 ". 84 "# 5&2 s
2
6
2 b6 s1RQ1 t
1
15 ¿ ¿ 1 & 2
2944> - au.o $%aFa a (( &2 cu"s.a arr%ba por u- ca&%-o r"c.o cu3o p"r*%+
$"r.%ca+ s" pu"!" apro=%&ar co- +a "cuac%/- &os.ra!a0 Cua-!o +a coor!"-a!a or%;o-.a+ !"+ au.o "s =14((&0 9Cu+"s so- +as co&po-"-."s !" su ac"+"rac%/-<
√( ) ( ) 2
V =
dx + d" dt dt
( ) ( )
2
d" ( x ) dt 2 dx +¿ dt v = √ ¿
¿
( )
2
dx +¿ dt v = √ ¿
2
2
2 x
¿ 1 +¿ ¿ ¿ v
2
¿ 2
2 ( x )
¿ 1 +¿ √ ¿ dx v = ¿ dt
2
2 ( x )
¿ 1 +¿ √ ¿
dx 27.78 m / s = ¿ dt
(0.0003 )( 400 ) 2¿ ¿ ¿2 1+¿ √ ¿ dx 27.78 = ¿ dt
dx =27.01 =¿ vx dt
Cua-!o =14(( d" = 2 ( 0.0003 ) ( 400 ) ( 27.01 ) dt d" = 6.4824 dt
dvx dvx dx dvx dvx dx = . =¿ = . dt dt dx dt dx dt x
¿ ¿ 2¿ 1+¿ vx =v ¿ x
¿ ¿ 2¿ 1+¿ dvx −v = ¿ dx
2
2
2 ( x ) ¿ 1 +¿
3/ 2
¿ 2 dvx −4 v c x = ¿ dx
E$a+ua-!o =14(( 2
0.0003 ¿
( 400 )
¿ / 400 ¿ ¿ ¿ 2 ( 0.003 )¿ 1 +¿ ¿ −4 (27.01 )¿ 2
3 2
dvx =¿ dx dvx =(−0.003678 )( 27.01) dt ax =−0.09934 m / s
x (
dx ) dt
d" = 2 ¿ dt
( ) 2
d x dx dx x + ( 2 c ) 2 dt dt d t 2
d " =2 ¿ 2 d t
Cua-!o =14(( a" =2 ( 0.0003 ) ( 400 ) (−0.09934 ) + ( 27.01 ) ( 2 ) ( 0.003 ) ( 27.01 ) a" = 0.4392
v =27.01 i + 6.4824 % ( m / s ) ⃗ 2
m/s ¿ a =−0.09934 i + 0.4392 % ¿ ⃗ tan & =
0.4824 27.01
&=13.49 5 e^ t =cos &i + sen&%
e^ t =cos ( 13.49 ) i + sen (13.49 ) % e^ t =0.9724 i +0.2332 %
e^ n =cos ( 13.39 + 90 ) i + sen ( 13.49 + 90 ) % e^ n =−02332 i + 0.9724 %
a^ t =ax ( 0.9724 ) i + a" ( 0.2332 ) % =−0.09659 i + 0.102421 %
a^ t =ax (−0.2332 ) i+ a" ( 0.9724 ) % =0.02316 i + 0.42707 % 2
0.102421 ¿
¿
−0.09659 ¿ 2+¿ ¿ |at |=√ ¿ 2
0.023316 ¿
¿ 2 0.42707 ¿ = 0.4276 ¿ |an|= √ ¿
29442 Las coor!"-a!as car."s%a-as !" u- pu-.o >u" s" &u"$" "- "+ p+a-o
=3 so-: 2
3
x =20 + 4 t m * " =10 −t m
9Cu+ "s "+ ra!%o !" +a cur$a.ura %-s.a-.-"o !" +a .ra3"c.or%a "- . 1 @s< x ´ =8 t
2
" ´ =−3 t 2
v =8 t i −3 t %
|V |t 3 1 √ ( 8∗3 )2+(−3∗(3 )2)2 1 @80'&2s " ´ ´ =− 6 t
x ´ ´ =8
a =8 i−6 t −1
&= tan (
−27 24
) 1 (044@8
an =8 sen (−0.844153986 ) + 18cos (−0.844153986 ) 2
an =5.98 m / s
Para "+ ra!%o !" +a cur$a.ura s" u.%+%;a +a s%)u%"-." *or&u+a: 6=
|V |2 an
6=218 m
2944?- Fo$"- pa.%-a sobr" +a sup"r*%c%" !" co-cr".o !" u- ca-a + $acío !"scr%.o por +a "cuac%/- &os.ra!a0 E+ Fo$"- par." !" " =20 pie 3 +a &a)-%.u! !" su
$"+oc%!a! "s. !a!a por v =√ 2 ( 32.2 ) (20 − " ) pie / s 0 9Cu+"s so- +as co&po-"-."s -or&a+ 3 .a-)"-c%a+ !" su ac"+"rac%/- cua-!o ++")a a+ *o-!o<
√( ) ( ) 2
v=
2
d" dx + dt dt 1 2
1 2 2
v = 9 ( 20− ") = 9 ( 20−C x ) 9 = 8.025
=0.03 2 = x
( )
v=
√
( )
2
d" dx = 2 Cx dt dt
2
,
√ ( ( )) ( ) dx 2 Cx dt
2
2
1
2
dx + = 9 ( 20− " ) 2 dt
( ) ( ) √( ) 2
2
dx dx 4 C x + = dt dt 2
( ) 2
d " dx d x = 2 C + 2 Cx 2 2 dt dt dt
dx dt
2
1
( 4 C x + 1 ) = dx √ ( 4 C 2 x 2 + 1 )= 9 (20 −C x2 ) 2 dt 2
2
1 2 2
dx 9 ( 20 −C x ) = 1 dt 2 2 ( 4 C x +1 ) 2
1 2
d x = 2 dt
− 9Cx 1
dx 9 4 C x ( 20 −C x − 1 dt 2 2 (4 C x + 1 )2
( ) + ) 1
( 20 −C x ) (4 C x 2 2
2
2
1
2
2
)
2 2
( ) dx dt
A+ *%-a+ !"+ ca-a+ ."-"&os >u" dx ft = 9 √ 20 =35.89 dt x=0 s
( )
( )
2
d" d x =0 * 2 dt x=0 dt
( ) 2
d " 2 dt
=0
x =0
( )
dx =2 C dt x =0
2
=77.28 x =0
ft s
E+ -)u+o co- r"sp"c.o a+ "F" = &= tan
−1
( ) d" dx
=0
x =0
La co&po-"-." !" +a ac"+"rac%/- .a-)"-c%a+, co-s.a-."0 & + 90 (¿)i + sin ( & + 90 ) % cos ¿ e^ n=¿
at =0
por>u" +a $"+oc%!a! "s
