Mibvunzo yeMienzaniso neKukurukurirana kweExponential Decay
Kuora kweExponential chiitiko chechisikigo chinowanikwa muzvikamu zvakasiyana-siyana zvakaita sefizikisi, kemesitiri, biology, uye economics. Semuenzaniso wemasvomhu, kuora kweexponential kunotsanangura maitiro ayo huwandu hwakapihwa hunoderera zvichienderana nehuwandu hwahwo hwazvino. Mumasvomhu, kuora kweexponential kunotevera chimiro chakajairika:
\[ N(t) = N_0 e^{-\lambda t} \]
di mana:
– \( N(t) \) ndiyo huwandu hwasara panguva \(t \),
– \( N_0 \) ndiyo nhamba yekutanga,
– \( \lambda \) ndiyo nguva dzose yekuora (inowanzonzi chiyero chekuora),
– \(t \) inguva,
– \( e \) ndiyo hwaro hwe logarithm yechisikigo (inenge 2.718).
Muchinyorwa chino, tichakurukura mimwe mienzaniso yematambudziko ekuora kwe exponential pamwe chete netsananguro dzawo kuti tinzwisise pfungwa iyi zvakadzama.
Muenzaniso Mubvunzo 1: Kuora kweRadioactive
Mubvunzo:
Chinhu chine mwaranzi chine hupenyu hwehafu yemakore mashanu. Dai pakutanga paive nemagiramu zana echinhu ichi, chingave chakawanda sei chaizosara mushure memakore gumi nemashanu?
Kukurukurirana:
Kuora kwemwaranzi kunogona kuenzaniswa uchishandisa fomura yekuora kwe exponential. Hafu-hupenyu (\( t_{1/2} \)) inguva inodiwa kuti hafu yehuwandu hwezvinhu zvine mwaranzi iore. Zvinozivikanwa kuti \( t_{1/2} = 5 \) makore.
Kutanga tinofanira kutsvaga decay constant \( \lambda \) nefomura:
\[ \lambda = \frac{\ln 2}{t_{1/2}} \]
\[ \lambda = \frac{\ln 2}{5} \inenge 0.1386 \chinyorwa{ gore}^{-1} \]
Saka, fomura yekuora kwe exponential ndeiyi:
\[ N(t) = N_0 e^{-\lambda t} \]
\[ N(t) = 100 e^{-0.1386 \kawa 15} \]
Iye zvino, tinoverenga kukosha:
\[ N(t) = 100 e^{-2.079} \]
\[ N(t) = 100 \kawa 0.125 \]
\[ N(t) \inenge 12.5 \text{ grams} \]
Saka, mushure memakore gumi nemashanu, anenge magiramu 12.5 ezvinhu zvine mwaranzi zvinosara.
Muenzaniso 2: Kuora kweCapacitor
Mubvunzo:
Capacitor ine chaji yekutanga \( Q_0 = 200 \text{C} \) inobvumirwa kuburitsa mucircuit. Nguva isingachinji ndeye \( \tau = 4 \text{s} \). Chaji yakawanda sei inosara mushure memasekonzi gumi?
Kukurukurirana:
Kana capacitor charge decay yaora, exponential model inoshandiswa ndeiyi:
\[ Q(t) = Q_0 e^{-t/\tau} \]
Zvapiwa \( Q_0 = 200 \text{ C} \) uye \( \tau = 4 \text{ s} \). Tinofanira kutsvaga \( Q(10) \):
\[ Q(10) = 200 e^{-10/4} \]
\[ Q(10) = 200 e^{-2.5} \]
Kuverenga huwandu hwe exponential:
\[ Q(10) = 200 \kawa 0.0821 \]
\[ Q(10) \inenge 16.42 \chinyorwa{ C} \]
Saka, mushure memasekonzi gumi, chaji yasara pa capacitor inenge iri 16.42 C.
Muenzaniso Mubvunzo 3: Kuora kwemakemikari
Mubvunzo:
Kemikari ine chiyero chekuora che \( \lambda = 0.05 \text{ days}^{-1} \). Zvinotora nguva yakareba sei kuti kemikari idzike kusvika pa25% yehuwandu hwayo hwepakutanga?
Kukurukurirana:
Tinotanga nefomura yakajairika yekuora kwe exponential:
\[ N(t) = N_0 e^{-\lambda t} \]
Tinoda kuti N(t) ive 25% ye \( N_0 \), kuitira kuti:
\[ 0.25 N_0 = N_0 e^{-0.05 t} \]
Kubvisa \( N_0 \) kubva kumativi ese:
\[ 0.25 = e^{-0.05 t} \]
Kushandisa ma logarithms echisikigo kugadzirisa nyaya dze exponential:
\[ \ln 0.25 = -0.05 t \]
\[ -1.3863 = -0.05 t \]
Kugadzirisa \(t \):
\[t = \frac{1.3863}{0.05} \]
\[t \inenge 27.726 \text{ days} \]
Saka, nguva inodiwa kuti kemikari idzike kusvika pa25% yehuwandu hwayo hwekutanga ingangoita mazuva 27.726.
Muenzaniso Mubvunzo 4: Kuora Kwehuwandu hweMabhakitiriya
Mubvunzo:
Huwandu hwemabhakitiriya hunoderera nekukurumidza zvekuti mushure memaawa matatu, huwandu hwevanhu hunosvika hafu yehuwandu hwekutanga. Kana huwandu hwepakutanga hwaive mabhakitiriya 8000, mabhakitiriya mangani anosara mushure memaawa mapfumbamwe?
Kukurukurirana:
Zvinozivikanwa kuti hafu yehupenyu \( t_{1/2} = 3 \) maawa. Kutanga tinowana kuora kunogara \( \lambda \):
\[ \lambda = \frac{\ln 2}{t_{1/2}} \]
\[ \lambda = \frac{\ln 2}{3} \inenge 0.231 \text{ hour}^{-1} \]
Mushure meizvozvo, tinoshandisa fomura yekuora kwe exponential:
\[ N(t) = N_0 e^{-\lambda t} \]
\[ N(9) = 8000 e^{-0.231 \kawa 9} \]
Kuverenga huwandu hwe exponential:
\[ N(9) = 8000 e^{-2.079} \]
\[ N(9) = 8000 \kawa 0.125 \]
\[ N(9) \inenge 1000 \]
Saka, mushure memaawa mapfumbamwe, mabhakitiriya angangoita chiuru acharamba aripo.
Mhedziso
Muenzaniso wekuora kwe exponential unopa nzira inoshanda yekugadzirisa matambudziko ane chekuita nekuora mumabasa akasiyana-siyana esainzi neeinjiniya. Nekunzwisisa pfungwa huru dzakadai sezvisingachinji zvekuora, hafu yehupenyu, uye kushandiswa kwemafomula e exponential, tinogona kuverenga shanduko muhuwandu nekufamba kwenguva zviri nyore. Matambudziko ekudzidzira akakurukurwa pamusoro apa anofanira kutibatsira kunzwisisa nekushandisa pfungwa yekuora kwe exponential mumamiriro ezvinhu akaoma.
