Motsin faɗuwa kyauta

Labari game da motsi na 'Yancin Faɗuwa' na 'Yanci

A rayuwar yau da kullum, sau da yawa muna ganin abubuwa da ke fuskantar motsi na faɗuwa kyauta, misali, motsi na 'ya'yan itace da ke faɗuwa daga bishiya, motsi na abubuwan da ke faɗuwa ko kuma suka faɗi daga wani tsayi. Me yasa abubuwa ke fuskantar motsi na faɗuwa kyauta? Idan aka lura da kallo, abu yana fuskantar faɗuwa kyauta kamar yana da madaidaicin gudu, ko kuma a wata ma'anar abu ba ya hanzartawa. Gaskiyar da ke faruwa, kowane abu da ya faɗi kyauta yana fuskantar hanzari akai-akai. Wannan dalili yana haifar da motsi na faɗuwa kyauta, gami da misalin motsi na layi mara daidaito. Ta yaya za a tabbatar da cewa abubuwa da ke fuskantar juyawar kyauta suna fuskantar hanzari akai-akai ko ƙaruwar saurinsa?

A saka ƙusa biyu a ƙasa sannan a jefa dutse daga tsayi daban-daban a kan kowace ƙusa. Za ku ga cewa ƙusa da aka yi wa duwatsu a matsayi mafi girma sun fi sauran ƙusa zurfi. Da girman matsayin dutsen daga saman ƙasa, haka nan girman gudun dutsen lokacin da ya bugi ƙasa har ya matse ƙusa sosai.

A da, motsin abubuwa da suka faɗi ƙasa wani batu ne mai ban sha'awa a falsafar halitta. Aristotle, wani masanin falsafa, ya taɓa cewa abu mai girma yana faɗuwa da sauri fiye da abubuwa masu sauƙi. Ra'ayin Aristotle ya rinjayi ra'ayoyin mutanen da suka rayu kafin zamanin Galileo, waɗanda suka yi la'akari da cewa abubuwa masu girma suna faɗuwa da sauri fiye da abubuwa masu sauƙi kuma saurin abubuwan da suka faɗi yana daidai da nauyin abu. Wataƙila kafin mu yi nazarin wannan batu, kai ma kana tunanin haka. Misali, muna faɗuwa da takarda da dutse daga tsayi ɗaya. Sakamakon da muka lura ya nuna cewa dutsen ya fara taɓa ƙasa ko ƙasa fiye da takarda. Yanzu, bari mu faɗuwa da duwatsu biyu daga tsayi ɗaya, inda dutse ɗaya ya fi ɗayan girma. Duwatsu biyu sun taɓa saman ƙasa a lokaci guda, idan aka kwatanta da dutse da takarda da muka faɗu a baya. Haka kuma za mu iya gwada faɗuwa da duwatsu da takarda a cikin nau'in dunƙule.

Me ke shafar motsin duwatsu ko takarda da ke faɗuwa? Ƙarfin gogayya ta iska! Juriyar iska ko gogayya tana shafar motsin faɗuwa ta 'free-fall' sosai. Galileo ya yi hasashen cewa duk abubuwa za su faɗi da sauri iri ɗaya idan babu iska ko wasu cikas. Galileo ya yi iƙirarin cewa duk abubuwa, masu nauyi ko masu sauƙi, suna faɗuwa da sauri iri ɗaya, aƙalla idan babu iska. Galileo ya yi imanin cewa iska tana aiki a matsayin cikas ga abubuwa masu haske waɗanda ke da babban saman. Amma a yanayi da yawa, ana iya yin watsi da wannan juriyar iska. A cikin ɗaki inda aka tsotse iska (vacuum), abubuwa masu haske kamar takarda da aka riƙe a kwance za su faɗi da sauri iri ɗaya da sauran abubuwa. Gudunmawar Galileo ga fahimtarmu game da motsin abubuwa masu faɗuwa za a iya taƙaita ta kamar haka:

A wani wuri a duniya kuma ba shi da juriyar iska, duk abubuwa suna faɗuwa da sauri iri ɗaya. Muna kiran wannan hanzarin da nauyi ya haifar kuma muna ba shi alama g. Girman g shine kimanin 9.8 m/s ^2. A cikin raka'o'in Tsarin Ingilishi, girman g shine kimanin ƙafa 32/s ^2. Alkiblar hanzarta nauyi tana zuwa tsakiyar duniya.

Ma'anar motsi na faɗuwar 'free fall'

Ana cewa abu yana fuskantar faɗuwa kyauta idan abu ya motsa a tsaye zuwa tsakiyar duniya kuma yayin motsi, abu yana fuskantar saurin nauyi akai-akai. Idan faɗuwa kyauta kusa da saman duniya, to abu yana fuskantar saurin nauyi akai-akai na 9.8 m/s ²

da kuma alkiblar hanzarta nauyi zuwa tsakiyar duniya (wanda ke daidai da saman duniya). Don sauƙaƙa lissafin, g shine 10 m/s ².

Akwai yanayi uku daban-daban:

1. Abubuwa suna motsawa a tsaye ƙasa ba tare da saurin farko ba (babu v _o ). Misali, 'ya'yan itacen suna faɗuwa daga bishiyar bayan an sake su daga tushe. Alkiblar motsi koyaushe tana ƙasa kuma abu yana fuskantar hanzari don haka g koyaushe yana da kyau. A cikin wasu littattafan kimiyyar lissafi da ake kira motsi na faɗuwa kyauta.

2. Abu yana motsawa a tsaye ƙasa da saurin farko (akwai v _o ). Misali, dutse da aka jefa a tsaye ƙasa. Alkiblar motsi koyaushe tana ƙasa kuma abu yana fuskantar hanzari don haka g koyaushe yana da kyau. A wasu littattafan kimiyyar lissafi, ana kiransa motsi a tsaye ƙasa.

3. Abu yana motsawa a tsaye sama da saurin farko, bayan ya kai matsakaicin tsayi, abu yana komawa ƙasa. A ce ka jefa duwatsu masu tsayi sama ka sake kama su lokacin da duwatsun suka motsa ƙasa. Lokacin da suke motsawa sama, abubuwa suna fuskantar raguwar gudu (gami da g), lokacin da suke motsawa a tsaye ƙasa, abu yana ƙaruwa (gami da g). A wasu littattafan kimiyyar lissafi, ana kiransa motsi a tsaye sama. Ya kamata a lura cewa idan abu ya fuskanci ɗaya daga cikin yanayi uku da ke sama, ana cewa abu yana yin motsi a lokacin faɗuwa.

Daidaiton motsi na faɗuwar 'free fall'

Motsin faɗuwa na 'free-fall' misali ne na motsi na layi mara tsari, saboda haka daidaiton motsi na 'free-fall' a zahiri iri ɗaya ne da daidaiton motsi na layi mara tsari kuma an daidaita shi da yanayin motsi na 'free-fall'.

h = tsayi (mita), v _o = saurin farko (mita/daƙiya), v _t = gudun ƙarshe (mita/daƙiya), t = lokaci (daƙiya), g = hanzarta nauyi (mita/daƙiya) = 9.8 m/s ² ko 10 m/s ².

Saurin nauyi yana nan a 10 m/s ² (g positive, abu yana tafiya ƙasa) yana nufin cewa saurin abu yana ƙaruwa da m/s 10 a kowace daƙiƙa 1. Bayan daƙiƙa 2, saurin abu yana ƙaruwa da m/s 20. Bayan daƙiƙa 3, saurin abu yana ƙaruwa da m/s 30. Saurin nauyi akai-akai shine m/s 10 2 ⁽ g positive, abu yana motsawa sama) yana nufin cewa saurin abu yana raguwa da m/s 10 a kowace daƙiƙa 1. Bayan daƙiƙa 2, saurin abu yana raguwa da m/s 20. Bayan daƙiƙa 3, saurin abu yana raguwa da m/s 30. Saurin gaggawa akai-akai ko raguwa akai-akai yana faruwa ne kawai kusa da saman duniya.

Misalin matsala ta 1:

Ana sakin mangwaro ya faɗi ƙasa. Idan wurin farko ya kasance mita 10 daga saman ƙasa kuma nauyin mangwaro ya kai gram 5, a tantance:

(a) saurin mangwaro idan ya isa ƙasa

(b) lokacin da mangwaro zai isa ƙasa.

g = 9.8 m/s ²

Magani:

Sananne: h = 10 m, g = 9.8 m/s ²

a) Saurin manga idan ya isa ƙasa

Ana ƙididdige taro a cikin lissafin motsi na 'free-fall'

b) Tazarar lokaci a cikin iska

Misalin matsala ta 2:

Ana sauke wani abu daga wani tsayi. A tantance:

(a) girman hanzarin abu

(b) nisan da abin ya yi a cikin daƙiƙa 2 na farko

(c) saurin abin bayan faɗuwar mita 50

(d) tsawon lokacin da ake buƙata don abin ya kai gudun mita 20/s,

(e) tsawon lokacin da ake buƙata don abin ya faɗi har zuwa mita 100

Magani:

Sananne: g = 9.8 m/s ²

a) Girman hanzarin abu

Hanzarta abu = hanzarta nauyi = g = 9.8 m/s ²

b) Nisan da abin ya yi tafiya a cikin daƙiƙa 2 na farko

Sananne: g = 9.8 m/s ² , t = 2 daƙiƙa

Ana so: h

h = 1⁄2 gt ² = 1⁄2 (9.8)(2) ² = (4.9)(4) = mita 19.6

c) Saurin abin bayan faɗuwa har zuwa mita 50

Sananne: h = 50 m, g = 9,8 m/s ²

Ana so: v _t

d) Nawa lokaci ake buƙata don abubuwa su kai gudun mita 20/s

Sananne: v _t = 20 m/s, g = 9,8 m/s ²

Ana so: t

e) Tazarar lokacin da ake buƙata don abubuwa su faɗi har zuwa mita 100

Ana so: h = 100 m, g = 9,8 m/s ²

Magani: t

Misalin matsala ta 3:

Ana jefa dutse a cikin rijiya mai saurin farko na 5 m/s. Idan dutsen ya faɗi cikin ruwa bayan daƙiƙa 4, a tantance:

(a) saurin dutsen idan ya shiga ruwa

(b) zurfin rijiyar

Magani:

a) Saurin dutse idan ya shiga ruwa

Wanda aka sani:

v _o = 5 m/s, t = 4 s, g = 9.8 m/s ²

Ana so: v _t

v _t = v _o + g t

v _t = 5 m/s + (9.8 m/s ² )(4 s) = 5 m/s + 39.2 m/s

v _t = 44.2 m/s

b) Zurfin rijiya

Wanda aka sani:

v _o = 5 m/s, t = 4 s, g = 9.8 m/s ²

Ana so: h

h = v _o t + ½ gt ²

h = (5)(4) + ½ (9.8)(4) ²

h = 20 + (4.9)(16)

h = 20 + 78.4

h = mita 98.4

Misalin matsala ta 4:

Daga saman ginin har zuwa mita 50, ana jefa fakiti a tsaye ƙasa da gudun mita 10/s. A tantance:

(a) Lokacin da ake cikin iska

(b) Saurin kunshin lokacin da ya bugi ƙasa

Magani:

a) Lokacin iska

Wanda aka sani:

h = 50 m, g = 9.8 m/s ² , v _o = 10 m/s

Ana so: t

h = v _o t + ½ gt ²

50 = 10 t + ½ (9.8) t ²

50 = 10 t + 4.9 t ²

4.9 t ² + 10 t – 50 = 0

Yi amfani da dabarar quadratic:

Lokacin da ake amfani da shi a iska = daƙiƙa 2.3

b) Saurin kunshin lokacin da ya bugi ƙasa

Wanda aka sani:

h = 50 m, g = 9.8 m/s ² , v _o = 10 m/s

Ana so: v _t

Misalin matsala ta 5:

Ana jefa ƙwallon a tsaye sama da saurin farko na 20 m/s. A ƙayyade matsakaicin tsayin da ƙwallon ta kai.

Magani:

Adadin vector wanda alkiblarsa ta sama tana da kyau, adadin vector wanda alkiblarsa ta ƙasa tana da kyau. Matsayin farko na ƙwallon an zaɓi shi azaman wurin tunani.

Wanda aka sani:

v _o = 20 m/s (alkiblar saurin farko yana sama, ana jefa ƙwallon sama, don haka v _o yana da kyau)

v _t = 0 m/s (gudun ƙwallon da ke matsakaicin tsayi shine 0 m/s)

g = – 9.8 m/s ² (alkiblar hanzarin nauyi yana ƙasa, don haka g yana da korau)

Ana so: h

Misalin matsala ta 6:

Ana jefa marmara a tsaye daga saman gini mita 100 sama da ƙasa a saurin farko na mita 20/s. A ƙayyade:

(a) Lokacin iska

(b) saurin marmara lokacin da ta bugi ƙasa

Magani:

Matsayin da aka jefa marmara a matsayin wurin tunani; saman ginin wurin tunani ne. Girman vector wanda alkiblarsa ta sama tana da kyau, girman vector wanda alkiblarsa ta ƙasa tana da korau.

a) Lokacin iska

Wanda aka sani:

h = – 100 m (h yana da korau saboda saman ƙasa yana ƙasa da matsayin farko ko wurin tunani)

v _o = 20 m/s (alkiblar saurin farko yana sama, don haka v _o yana da kyau)

g = – 9.8 m/s ² (alkiblar hanzarin nauyi yana ƙasa, don haka g yana da korau)

Ana so: t

Yi amfani da dabarar quadratic:

Lokacin da ake buƙata don isa ƙasa tun daga lokacin

an jefa ƙwallon = daƙiƙa 7.

b) Saurin marmara idan ya bugi saman ƙasa

Wanda aka sani:

h = -100 m (mara kyau saboda saman ƙasa yana ƙasa da wurin tunani ko matsayin farko)

v _o = 20 m/s (alkiblar saurin farko yana sama, don haka v _o yana da kyau)

g = -9.8 m/s ² (alkiblar hanzarin nauyi yana ƙasa, don haka g yana da korau).

Ana so: v _t

Tambayoyi da amsoshi game da Free fall motion

1. Menene motsi na faɗuwa kyauta?

   Motsin faɗuwa kyauta shine motsin abu a ƙarƙashin tasirin nauyi kaɗai, ba tare da wani ƙarfi da ke aiki a kansa ba, kamar juriyar iska.

2. Menene saurin gudu saboda nauyi?

   Saurin da abu ke samu sakamakon nauyi shine saurin da abu ke samu lokacin da yake faɗuwa kusa da saman Duniya. Yawanci ana nuna shi a matsayin 'g' kuma yana da kimanin ƙimar 9.8 m/s².

3. Menene dabarar ƙaura a cikin motsi na faɗuwa kyauta, idan aka yi la'akari da saurin farko da lokacin?

   Tsarin shine h = ut + ½gt², inda h shine motsi (tsawo), u shine saurin farko, g shine hanzari saboda nauyi, kuma t shine lokaci.

4. Ta yaya saurin abu ke canzawa yayin motsi na faɗuwa kyauta?

   A lokacin faɗuwar 'free fall', saurin abu yana ƙaruwa a layi ɗaya akan lokaci saboda saurin nauyi akai-akai.

5. Menene dabarar gudu na ƙarshe a cikin motsi na faɗuwa kyauta, idan aka yi la'akari da saurin farko da lokaci?

   Tsarin shine v = u + gt, inda v shine saurin ƙarshe, u shine saurin farko, kuma g shine hanzari saboda nauyi.

6. Me zai faru da saurin abu a lokacin da yake kan hanyarsa a lokacin da yake faɗuwa kyauta?

   A lokacin da yake kan kololuwar hanyarsa, saurin abin ya zama sifili na ɗan lokaci.

7. Menene alamar da ba ta da kyau ke wakilta a cikin mahallin motsi na faɗuwa kyauta?

   Alamar mara kyau yawanci tana nuna alkiblar da ke akasin alkiblar da aka zaɓa mai kyau. Dangane da mahallin, tana iya nuna faɗuwa ƙasa (idan sama ta kasance mai kyau) ko kuma jefa ƙasa (idan sama ta kasance mara kyau).

8. Me ke faruwa da hanzarin abu a lokacin faɗuwar 'free fall' lokacin da ya kai matsakaicin tsayinsa?

   Hanzarta abu a matsakaicin tsayinsa a lokacin faɗuwar 'free fall' ya kasance daidai da g (kimanin -9.8 m/s² kusa da saman Duniya), yana nuna ƙasa.

9. Ta yaya juriyar iska ke shafar motsin faɗuwa kyauta?

   A zahiri, juriyar iska na iya rage saurin saukowa abu sosai, wanda hakan ke sa motsi ba ya zama faɗuwa kyauta. Duk da haka, a cikin matsalolin kimiyyar lissafi da yawa, ana yin watsi da juriyar iska saboda sauƙi.

10. Menene lokacin tashi a cikin motsi na faɗuwar kyauta?

Lokacin tashi shine jimlar lokacin da abin ya shafe a cikin iska. Ga wani abu da aka harba kuma ya sauka a tsayi ɗaya, ana iya samun lokacin tashi ta hanyar dabarar t = 2u/g.

Matsaloli da mafita game da Free fall motion

1. matsala: An sauke dutse daga wani dutse mai tsayin mita 78.4. Tsawon wane lokaci dutsen zai ɗauka kafin ya isa ƙasa? Magani: Muna amfani da lissafin h = ½gt². Idan muka warware lokaci, t = √(2h/g) = √(2×78.4/9.8) = 4 s.
2. matsala: Ana jefa ƙwallon sama da saurin farko na 19.6 m/s. Har yaushe ƙwallon take tafiya? Magani: Ta amfani da lissafin h = v₀t – ½gt² a matsakaicin tsayi (inda saurin ƙarshe shine 0), tsayin h = v₀² / (2g) = (19.6)² / (2×9.8) = 20 m.
3. matsala: Ana jifa dutse sama da saurin farko na m10/s. Yaya saurinsa zai kasance bayan daƙiƙa 2? Magani: Lissafin v = v₀ – gt yana ba da saurin v = 10 – 9.8×2 = -9.6 m/s.
4. matsala: An jefar da tsabar kuɗi a cikin rijiya kuma ta bugi ruwa bayan daƙiƙa 3. Yaya zurfin rijiyar yake? Magani: Ta amfani da h = ½gt², zurfin h = ½x9.8×3² = 44.1 m.
5. matsala: Littafi ya faɗo daga kan teburi ya faɗi ƙasa bayan daƙiƙa 0.5. Tsawon teburin nawa ne? Magani: Ta amfani da h = ½gt², tsayin h = ½x9.8x(0.5)² = 1.225 m.
6. matsala: Ana jefa ƙwallon sama da saurin farko na m20/s. Yaushe zai kai matsakaicin tsayinsa? Magani: A matsakaicin tsayi, v = 0. Magance t = (v – v₀) / -g, lokacin t = (0 – 20) / -9.8 = 2.04 s.
7. matsala: Abu yana faɗuwa na tsawon daƙiƙa 6. Menene saurinsa na ƙarshe? Magani: Ta amfani da v = v₀ + gt tare da saurin farko v₀ = 0, saurin ƙarshe v = 0 + 9.8×6 = 58.8 m/s.
8. matsala: Tuffa ta faɗo daga bishiya kuma tana ɗaukar daƙiƙa 1.5 kafin ta faɗi ƙasa. Tsawon bishiyar ya kai nawa? Magani: Ta amfani da h = ½gt², tsayin h = ½x9.8x(1.5)² = 11 m.
9. matsala: Ana harba ƙwallon ƙafa a tsaye sama da saurin farko na m25/s. Tsawon lokacin har sai ta faɗi ƙasa? Magani: Lokaci zuwa matsakaicin tsayi t = v₀ / g = 25 / 9.8 = 2.55 s. Jimlar lokacin da za a buga ƙasa sau biyu ne, don haka t = 2×2.55 = 5.1 s.
10. matsala: An jefar da dutse daga kan gada ya bugi ruwa bayan daƙiƙa 4. Tsawon gadar nawa ne? Magani: Ta amfani da h = ½gt², tsayin h = ½x9.8×4² = 78.4 m.
11. matsala: Ana harba roka kai tsaye a gudun mita 50/s. Yaya tsayinsa yake? Magani: Ta amfani da h = v₀² / (2g), tsayin h = (50)² / (2×9.8) = 127.55 m.
12. matsala: Ana jefa ƙwallon ƙasa da saurin farko na m10/s. Menene saurinta bayan daƙiƙa 2? Magani: Ta amfani da v = v₀ + gt, saurin v = 10 + 9.8×2 = 29.6 m/s.
13. matsala: Ana jefa ƙwallon a tsaye sama kuma ta koma ƙasa cikin daƙiƙa 6. Menene saurinta na farko? Magani: Ta amfani da v₀ = gt / 2 (tunda jimlar lokaci sau biyu ne lokacin da za a kai ga matsakaicin tsayi), saurin farko v₀ = 9.8×6 / 2 = 29.4 m/s.
14. matsala: Abu yana faɗuwa na tsawon daƙiƙa 10. Har yaushe yake faɗuwa? Magani: Ta amfani da h = ½gt², nisan h = ½x9.8×10² = 490 m.
15. matsala: An jefar da marmara daga hasumiya kuma yana ɗaukar daƙiƙa 5 kafin ya faɗi ƙasa. Tsawon hasumiyar nawa ne? Magani: Ta amfani da h = ½gt², tsayin h = ½x9.8×5² = 122.5 m.
16. matsala: Ana jefa ƙwallon baseball a tsaye sama da gudu na farko na m15/s. Yaushe zai kai matsakaicin tsayinsa? Magani: A matsakaicin tsayi, v = 0. Magance t = (v – v₀) / -g, lokacin t = (0 – 15) / -9.8 = 1.53 s.
17. matsala: Ana jefar da abu ya faɗi na tsawon daƙiƙa 7. Menene saurinsa na ƙarshe? Magani: Ta amfani da v = v₀ + gt tare da saurin farko v₀ = 0, saurin ƙarshe v = 0 + 9.8×7 = 68.6 m/s.
18. matsala: Ana jefa dutse sama da saurin farko na m30/s. Yaya saurinsa zai kasance bayan daƙiƙa 3? Magani: Ta amfani da v = v₀ – gt, saurin v = 30 – 9.8×3 = 0.6 m/s.
19. matsala: An jefar da dutse daga wani dutse ya faɗi ƙasa bayan daƙiƙa 8. Tsawon dutsen nawa ne? Magani: Ta amfani da h = ½gt², tsayin h = ½x9.8×8² = 313.6 m.

20. Matsala: Ana harba kibiya a miƙe a gudun mita 60/s. Tsawon lokacin da zai ɗauka har sai ta faɗi ƙasa? Magani: Lokaci zuwa matsakaicin tsayi t = v₀ / g = 60 / 9.8 = 6.12 s. Jimlar lokacin da za a buga ƙasa sau biyu ne, don haka t = 2×6.12 = 12.24 s.