Mafunso ndi Zitsanzo za Kuwonongeka kwa Exponential
Kuwonongeka kwa exponential ndi chinthu chachilengedwe chomwe chimapezeka m'magawo osiyanasiyana monga fizikisi, chemistry, biology, ndi zachuma. Monga chitsanzo cha masamu, kuwonongeka kwa exponential kumafotokoza njira yomwe kuchuluka koperekedwa kumachepera molingana ndi kuchuluka kwake kwamakono. Mu masamu, kuwonongeka kwa exponential kumatsatira mawonekedwe onse:
\[ N(t) = N_0 e^{-\lambda t} \]
Kumene:
– \( N(t) \) ndi kuchuluka komwe kwatsala panthawiyo \(t \),
– \( N_0 \) ndi nambala yoyambira,
– \( \lambda \) ndi nthawi yokhazikika yowola (nthawi zambiri imatchedwa kuchuluka kwa kuwola),
– \(t \) ndi nthawi,
– \( e \) ndiye maziko a logarithm yachilengedwe (pafupifupi 2.718).
M'nkhaniyi, tikambirana zitsanzo zina za mavuto a kuwonongeka kwa ma exponential pamodzi ndi njira zawo zothetsera vutoli kuti timvetse bwino mfundo imeneyi.
Chitsanzo Funso 1: Kuwonongeka kwa Ma Radioactive
Funso:
Mankhwala owopsa amakhala ndi theka la moyo wa zaka 5. Ngati poyamba panali magalamu 100 a mankhwalawa, kodi angatsale angati patatha zaka 15?
Kukambirana:
Kuwonongeka kwa ma radiation kumatha kupangidwa pogwiritsa ntchito njira ya exponential decay. Hafu ya moyo (\( t_{1/2} \)) ndi nthawi yomwe imafunika kuti theka la kuchuluka kwa zinthu zowopsa ziwole. Amadziwika kuti \( t_{1/2} = 5 \) zaka.
Choyamba tiyenera kupeza nthawi zonse ya decay \( \lambda \) ndi fomula iyi:
\[ \lambda = \frac{\ln 2}{t_{1/2}} \]
\[ \lambda = \frac{\ln 2}{5} \pafupifupi 0.1386 \text{ year}^{-1} \]
Motero, njira ya exponential decay ndi iyi:
\[ N(t) = N_0 e^{-\lambda t} \]
\[ N(t) = 100 e^{-0.1386 \nthawi 15} \]
Tsopano, tikuwerengera mtengo:
\[ N(t) = 100 e^{-2.079} \]
\[ N(t) = 100 \nthawi 0.125 \]
\[ N(t) \pafupifupi 12.5 \malemba{ magalamu} \]
Choncho, patatha zaka 15, pafupifupi magalamu 12.5 a zinthu zowononga ma radiation atsala.
Chitsanzo 2: Kuwonongeka kwa Capacitor
Funso:
Kachipangizo kokhala ndi mphamvu yoyambira \( Q_0 = 200 \text{C} \) kamaloledwa kutulutsa mphamvu mu circuit. Nthawi yokhazikika ndi \( \tau = 4 \text{s} \). Kodi mphamvu yotsala ndi yotani pakatha masekondi 10?
Kukambirana:
Pankhani ya kuwonongeka kwa mphamvu ya capacitor, chitsanzo cha exponential chomwe chimagwiritsidwa ntchito ndi:
\[ Q(t) = Q_0 e^{-t/\tau} \]
Titapatsidwa \( Q_0 = 200 \text{ C} \) ndi \( \tau = 4 \text{ s} \). Tifunika kupeza \( Q(10) \):
\[ Q(10) = 200 e^{-10/4} \]
\[ Q(10) = 200 e^{-2.5} \]
Kuwerengera ma exponential values:
\[ Q(10) = 200 \nthawi 0.0821 \]
\[ Q(10) \pafupifupi 16.42 \malemba{ C} \]
Kotero, patatha masekondi 10, mphamvu yotsala pa capacitor ndi pafupifupi 16.42 C.
Chitsanzo Funso 3: Kuwonongeka kwa Mankhwala
Funso:
Mankhwala ali ndi nthawi yofanana ndi ya \( \lambda = 0.05 \text{ days}^{-1} \). Kodi zimatenga nthawi yayitali bwanji kuti mankhwalawo atsike kufika pa 25% ya kuchuluka kwake koyambirira?
Kukambirana:
Timayamba ndi njira yonse ya kuwonongeka kwa exponential:
\[ N(t) = N_0 e^{-\lambda t} \]
Tikufuna kuti N(t) ikhale 25% ya \( N_0 \), kotero kuti:
\[ 0.25 N_0 = N_0 e^{-0.05 t} \]
Kuchotsa \( N_0 \) mbali zonse ziwiri:
\[ 0.25 = e^{-0.05 t} \]
Kugwiritsa ntchito ma logarithm achilengedwe pothetsa milandu yodziwika bwino:
\[ \ln 0.25 = -0.05 t \]
\[ -1.3863 = -0.05 t \]
Kuthetsa \(t \):
\[ t = \frac{1.3863}{0.05} \]
\[t \pafupifupi 27.726 \malemba{masiku} \]
Choncho, nthawi yomwe mankhwalawo amafunika kuti achepetse kufika pa 25% ya kuchuluka kwake koyamba ndi pafupifupi masiku 27.726.
Chitsanzo Funso 4: Kuwola kwa Bakiteriya
Funso:
Chiwerengero cha mabakiteriya chimachepa pang'onopang'ono kotero kuti patatha maola atatu, chiwerengerocho chimakhala theka la chiwerengero chake choyamba. Ngati chiwerengero choyamba chinali mabakiteriya 8000, ndi mabakiteriya angati omwe amatsala patatha maola 9?
Kukambirana:
Zimadziwika kuti theka la moyo \( t_{1/2} = 3 \) maola. Choyamba timapeza nthawi zonse ya kuwonongeka \( \lambda \):
\[ \lambda = \frac{\ln 2}{t_{1/2}} \]
\[ \lambda = \frac{\ln 2}{3} \pafupifupi 0.231 \text{ ola}^{-1} \]
Pambuyo pake, timagwiritsa ntchito njira ya exponential decay:
\[ N(t) = N_0 e^{-\lambda t} \]
\[ N(9) = 8000 e^{-0.231 \nthawi 9} \]
Kuwerengera ma exponential values:
\[ N(9) = 8000 e^{-2.079} \]
\[ N(9) = 8000 \nthawi 0.125 \]
\[ N(9) \pafupifupi 1000 \]
Choncho, patatha maola 9, mabakiteriya pafupifupi 1000 adzatsala.
Mapeto
Chitsanzo cha kuwonongeka kwa exponential chimapereka njira yabwino yothetsera mavuto okhudzana ndi njira zowola m'njira zosiyanasiyana zasayansi ndi ukadaulo. Mwa kumvetsetsa mfundo zoyambira monga zokhazikika za kuwonongeka, theka la miyoyo, ndi kugwiritsa ntchito ma formula a exponential, titha kuwerengera kusintha kwa kuchuluka pakapita nthawi mosavuta. Mavuto a machitidwe omwe afotokozedwa pamwambapa ayenera kutithandiza kumvetsetsa ndikugwiritsa ntchito lingaliro la kuwonongeka kwa exponential m'mikhalidwe yovuta kwambiri.
