Răspuns :
1)Se da:
v1=60 km/h
v2=15 km/h
v3=45 km/h
d12=d
t12=t3=t
t1=t2=t/2
d3-?
vmed-?
v=d/t
dtot=d1+d2+d3=v1t1+v2t2+v3t3
t(tot)=t1+t2+t3
vmed=v1t1+v2t2+v3t3/t1+t2+t3
vmed=(60t/2 + 15t/2 +45t)/2t=60/4+15/4+45/2=
41,25( km/h)
PS:posibil sa nu fie corect,dar asta e din cauza neformulatii corecte a conditiei problemei
(pentru moderator:inainte a a sterge raspunsul,incercati dvs. sa intelegeti conditia)
2)se da:
v1=v
v2=1,5v1=1,5v
d1=d2
vmed=43,2km/h
Asa cum stim ca distantele sunt egale na dam seama ca d2 va fi parcurs de 1,5 ori mai incet,din cauza vitezei mai mari:
t2=1,5t1=1,5t
vmed=dtot/ttot=v1t1+v2t2/t1+t2
vmed=v1,5t+1,5vt/1,5t+t
vmed=3vt/2,5t=1,2v
v=v1
v2=vmed/1,2
v2=43,2/1,2=36 km/h
v1=1,5v2
v1=1,5*36=54 km/h
v1=60 km/h
v2=15 km/h
v3=45 km/h
d12=d
t12=t3=t
t1=t2=t/2
d3-?
vmed-?
v=d/t
dtot=d1+d2+d3=v1t1+v2t2+v3t3
t(tot)=t1+t2+t3
vmed=v1t1+v2t2+v3t3/t1+t2+t3
vmed=(60t/2 + 15t/2 +45t)/2t=60/4+15/4+45/2=
41,25( km/h)
PS:posibil sa nu fie corect,dar asta e din cauza neformulatii corecte a conditiei problemei
(pentru moderator:inainte a a sterge raspunsul,incercati dvs. sa intelegeti conditia)
2)se da:
v1=v
v2=1,5v1=1,5v
d1=d2
vmed=43,2km/h
Asa cum stim ca distantele sunt egale na dam seama ca d2 va fi parcurs de 1,5 ori mai incet,din cauza vitezei mai mari:
t2=1,5t1=1,5t
vmed=dtot/ttot=v1t1+v2t2/t1+t2
vmed=v1,5t+1,5vt/1,5t+t
vmed=3vt/2,5t=1,2v
v=v1
v2=vmed/1,2
v2=43,2/1,2=36 km/h
v1=1,5v2
v1=1,5*36=54 km/h
[tex] 1) \ Aflam\ viteza \ medie(v_m')\ pe \ a \ doua\ jumatate \ a \ drumului:[/tex]
[tex] d_2=d_1'+d_2' [/tex]
[tex]t_2=t_1'+t_2' [/tex]
[tex] t_1'=t_2'=\frac{t_2}{2} [/tex]
[tex] d_1'=v_2*t_1'=\frac{v_2*t_2}{2} [/tex]
[tex]d_2'=v_3*t_2'=\frac{v_2*t_2}{2} [/tex]
[tex] v_m'=\frac{d_2}{t_2}=\frac{d_1'+d_2'}{t_2}=\frac{\frac{v_2*t_2+v_3*t_2}{2}}{t_2}=\frac{v_2*t_2+v_3*t_2}{2t_2}=\frac{t_2(v_2+v_3)}{2t_2}=\frac{v_2+v_3}{2} [/tex] .
[tex]Aflam \ viteza \ medie(v_m) \ pe \ tot \ drumul :[/tex]
[tex] v_m=\frac{d}{t} [/tex]
[tex] d=d_1+d_2 [/tex]
[tex] d_1=d_2=\frac{d}{2} [/tex]
[tex] t=t_1+t_2 [/tex]
[tex] t_1=\frac{d_1}{v_1}=\frac{\frac{d}{2}}{v_1}=\frac{d}{2v_1} [/tex]
[tex] t_2=\frac{d_2}{v_m'}=\frac{\frac{d}{2}}{v_m'}=\frac{d}{2v_m'} [/tex]
[tex] v_m=\frac{d}{t}=\frac{d}{\frac{d}{2v_1}+\frac{d}{2v_m'}} [/tex]
[tex] v_m=\frac{d}{\frac{d(v1+v_m')}{2v1v_m'}}=\frac{2dv_1v_m'}{d(v_1+v_m')}=\frac{2v_1v_m'}{v_1+v_m'} \ (1) [/tex]
[tex]Inlocuim \ valoarea \ data \ lui \ v_m' \ in \ relatia \ (1) :[/tex]
[tex] v_m=\frac{2v_1\frac{v_2+v_3}{2}}{v_1+\frac{v_2+v_3}{2}}=\frac{\frac{2v_1(v2+v3)}{2}}{\frac{2v_1+v_2+v_3}{2}}=\frac{2*2v_1(v_2+v_3)}{2*(2v_1+v_2+v_3)}=\frac{2v_1(v_2+v_3)}{2v_1+v_2+v_3} =40 \frac{km}{h} [/tex]
[tex] 2) \ v_1=nv_2 [/tex]
[tex] d=d_1+d_2 [/tex]
[tex] d_1=d_2=\frac{d}{2} [/tex]
[tex] t=t_1+t_2 [/tex]
[tex] t_1=\frac{d_1}{v_1}=\frac{\frac{d}{2}}{v_1}=\frac{d}{2v_1} [/tex]
[tex] t_2=\frac{d_2}{v_2}=\frac{\frac{d}{2}}{v_2}=\frac{d}{2v_2} [/tex]
[tex] v_m=\frac{d}{t}=\frac{d}{\frac{d}{2v_1}+\frac{d}{2v_2}}=\frac{d}{\frac{d(v_1+v_2)}{2v_1v_2}}=\frac{2dv_1v_2}{d(v_1+v_2)}=\frac{2v_1v_2}{v_1+v_2} (1) [/tex]
[tex] Inlocuim \ valoarea \ data \ lui \ v_1 \ in \ relatia \ (1): [/tex]
[tex] v_m=\frac{2nv_2^2}{nv_2+v_2} [/tex]
[tex] v_m=\frac{2nv_2^2}{v_2(n+1)}=\frac{2nv_2}{n+1} [/tex]
[tex] v_m(n+1)=2nv_2 [/tex]
[tex] v_2=\frac{v_m(n+1)}{2n}=36 \frac{km}{h} [/tex]
[tex] v_1=nv_2=54 \frac{km}{h} [/tex]
[tex] Obs: \ Atasamentul \ este \ pentru \ ex \ 1 . [/tex]
[tex] d_2=d_1'+d_2' [/tex]
[tex]t_2=t_1'+t_2' [/tex]
[tex] t_1'=t_2'=\frac{t_2}{2} [/tex]
[tex] d_1'=v_2*t_1'=\frac{v_2*t_2}{2} [/tex]
[tex]d_2'=v_3*t_2'=\frac{v_2*t_2}{2} [/tex]
[tex] v_m'=\frac{d_2}{t_2}=\frac{d_1'+d_2'}{t_2}=\frac{\frac{v_2*t_2+v_3*t_2}{2}}{t_2}=\frac{v_2*t_2+v_3*t_2}{2t_2}=\frac{t_2(v_2+v_3)}{2t_2}=\frac{v_2+v_3}{2} [/tex] .
[tex]Aflam \ viteza \ medie(v_m) \ pe \ tot \ drumul :[/tex]
[tex] v_m=\frac{d}{t} [/tex]
[tex] d=d_1+d_2 [/tex]
[tex] d_1=d_2=\frac{d}{2} [/tex]
[tex] t=t_1+t_2 [/tex]
[tex] t_1=\frac{d_1}{v_1}=\frac{\frac{d}{2}}{v_1}=\frac{d}{2v_1} [/tex]
[tex] t_2=\frac{d_2}{v_m'}=\frac{\frac{d}{2}}{v_m'}=\frac{d}{2v_m'} [/tex]
[tex] v_m=\frac{d}{t}=\frac{d}{\frac{d}{2v_1}+\frac{d}{2v_m'}} [/tex]
[tex] v_m=\frac{d}{\frac{d(v1+v_m')}{2v1v_m'}}=\frac{2dv_1v_m'}{d(v_1+v_m')}=\frac{2v_1v_m'}{v_1+v_m'} \ (1) [/tex]
[tex]Inlocuim \ valoarea \ data \ lui \ v_m' \ in \ relatia \ (1) :[/tex]
[tex] v_m=\frac{2v_1\frac{v_2+v_3}{2}}{v_1+\frac{v_2+v_3}{2}}=\frac{\frac{2v_1(v2+v3)}{2}}{\frac{2v_1+v_2+v_3}{2}}=\frac{2*2v_1(v_2+v_3)}{2*(2v_1+v_2+v_3)}=\frac{2v_1(v_2+v_3)}{2v_1+v_2+v_3} =40 \frac{km}{h} [/tex]
[tex] 2) \ v_1=nv_2 [/tex]
[tex] d=d_1+d_2 [/tex]
[tex] d_1=d_2=\frac{d}{2} [/tex]
[tex] t=t_1+t_2 [/tex]
[tex] t_1=\frac{d_1}{v_1}=\frac{\frac{d}{2}}{v_1}=\frac{d}{2v_1} [/tex]
[tex] t_2=\frac{d_2}{v_2}=\frac{\frac{d}{2}}{v_2}=\frac{d}{2v_2} [/tex]
[tex] v_m=\frac{d}{t}=\frac{d}{\frac{d}{2v_1}+\frac{d}{2v_2}}=\frac{d}{\frac{d(v_1+v_2)}{2v_1v_2}}=\frac{2dv_1v_2}{d(v_1+v_2)}=\frac{2v_1v_2}{v_1+v_2} (1) [/tex]
[tex] Inlocuim \ valoarea \ data \ lui \ v_1 \ in \ relatia \ (1): [/tex]
[tex] v_m=\frac{2nv_2^2}{nv_2+v_2} [/tex]
[tex] v_m=\frac{2nv_2^2}{v_2(n+1)}=\frac{2nv_2}{n+1} [/tex]
[tex] v_m(n+1)=2nv_2 [/tex]
[tex] v_2=\frac{v_m(n+1)}{2n}=36 \frac{km}{h} [/tex]
[tex] v_1=nv_2=54 \frac{km}{h} [/tex]
[tex] Obs: \ Atasamentul \ este \ pentru \ ex \ 1 . [/tex]
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