Ngā Tauira Pātai e Matapaki ana i te Ture Tiaki Pūngao
Ko te Ture Tiaki Pūngao he mātāpono taketake i roto i te ahupūngao, e kī ana kāore e taea te waihanga, te whakangaro rānei i te pūngao i roto i tētahi pūnaha kati, engari ka taea anake te whakarerekē mai i tētahi āhua ki tētahi atu. He mea nui tēnei ariā, ā, he maha ngā wā e whakamahia ana i roto i ngā momo peka o te pūtaiao me te hangarau, tae atu ki te miihini, te thermodynamics, me te electromagnetism. I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira raruraru e pā ana ki te Ture Tiaki Pūngao, me ngā whakamārama taipitopito.
Tauira Pātai 1: Te Pūngao Ā-ringa i roto i tētahi Mea e Hinga Noa ana
Pātai: Ka taka tetahi pōro he 0,5 kg te taumaha mai i te teitei o te 20 mita. Kaua e aro ki te ātete hau. He aha te tere o te pōro ina tae ki te whenua?
Kōrero:
Hipanga 1: Tāutuhia te pūngao tīmatanga me te pūngao whakamutunga.
Ko te pūngao tīmatanga ina eke te pōro ki te teitei o te 20 mita ko te pūngao pūmanawa ā-papa (EP), ka taea te tatau mā te whakamahi i te tātai:
\[ EP = mgh \]
Dimana:
– Ko te taumaha o te pōro ko te \( m \) (0,5 kg)
– Ko te whakaterenga nā te kaha ā-papatipu te \( g \) (9,8 m/s²)
– Ko te teitei (20 m) ko te \( h \)
\[ EP_{tīmatanga} = 0,5 \times 9,8 \times 20 = 98 \, \text{J} \]
Ko te pūngao whakamutunga ina tae te pōro ki te whenua ko te pūngao nekeneke (EK), ka taea te tatau mā te whakamahi i te tātai:
\[ EK = \frac{1}{2} mv^2 \]
Dimana:
– Ko te tere o te pōro ina tae ki te whenua ko \( v \)
Hipanga 2: Whakamahia te Ture Tiaki Pūngao hei kī ko te pūngao pūmanawa tīmatanga he ōrite ki te pūngao nekeneke whakamutunga.
\[ EP_{tīmatanga} = EK_{whakamutunga} \]
\[ 98 = \frac{1}{2} \times 0,5 \times v^2 \]
Hipanga 3: Whakaotia te whārite hei kimi i te \( v \).
\[ 98 = 0,25v^2 \]
\[ v^2 = \frac{98}{0,25} = 392 \]
\[ v = \sqrt{392} \approx 19,8 \, \text{m/s} \]
Nō reira, ko te tere o te pōro ina tae ki te whenua he 19,8 m/s pea.
Tauira Pātai 2: Pūngao i roto i te Puna
Pātai: He puna me te pūmau puna \( k \) = 200 N/m e pēhia ana i te 0,1 mita mai i tōna tūranga taurite. E hia te kaha pūmanawa e rongoa ana i roto i te puna?
Kōrero:
Hipanga 1: Whakamahia te tātai pūngao pūmanawa rapa.
\[ EP_{pūngāwari} = \frac{1}{2} kx^2 \]
Dimana:
– Ko te pūmau puna ko te \( k \) (200 N/m)
– Ko te tawhiti kōpeketanga te \( x \) (0,1 m)
Hipanga 2: Monohia ngā uara e mōhiotia ana ki roto i te tātai.
\[ EP_{pūngao} = \frac{1}{2} \times 200 \times (0,1)^2 \]
\[ EP_{pūngao} = \frac{1}{2} \times 200 \times 0,01 \]
\[ EP_{pūngāwerewere} = 1 \, \text{J} \]
Nō reira, ko te pūngao pūmanawa rapa e rongoatia ana i roto i te puna ko 1 Joule.
Tauira Pātai 3: Te Pūngao Pūmanawa Ā-Toi me te Pūngao Kinetic i roto i te Nekehanga Parabolic
Pātai: Ka whakarewaina he pere he 2 kg te taumaha me te tere tīmatanga o te 30 m/s i te koki o te 45° ki te whakapae. He aha ngā pūngao nekeneke me ngā pūngao pūmanawa o te pere i te pūwāhi teitei rawa o tōna ara?
Kōrero:
Hipanga 1: Wehea te tere tīmatanga ki ngā wāhanga whakapae me ngā wāhanga poutū.
\[ v_{x} = v_0 \cos \theta \]
\[ v_{y} = v_0 \sin \theta \]
Dimana:
– Ko te tere tīmatanga (30 m/s) te \( v_0 \)
– Ko te koki whakarewatanga (45°) te \( \theta \)
\[ v_{x} = 30 \cos 45° = 30 \times \frac{\sqrt{2}}{2} = 21,21 \, \text{m/s} \]
\[ v_{y} = 30 \sin 45° = 30 \times \frac{\sqrt{2}}{2} = 21,21 \, \text{m/s} \]
Hipanga 2: I te pūwāhi teitei rawa, ko te tere poutū (v_y) he 0, engari ko te tere whakapae (v_x) ka noho pūmau tonu.
\[ v_{x \, pūwāhi \, teitei rawa} = 21,21 \, \text{m/s} \]
Hipanga 3: Tātaihia te pūngao nekeneke o te pere i tōna pūwāhi teitei rawa.
\[ EK = \frac{1}{2} mv^2 \]
\[ EK_{ira \, teitei rawa} = \frac{1}{2} \times 2 \times (21,21)^2 \]
\[ EK_{ira \, teitei rawa} = 1 \times 449,21 = 449,21 \, \text{J} \]
Hipanga 4: Tātaihia te teitei mōrahi i taea e te pere.
\[ h = \frac{v_{y}^2}{2g} \]
\[ h = \frac{(21,21)^2}{2 \times 9,8} \]
\[ h \tata ki te 22,9 \, \kuputuhi{m} \]
Hipanga 5: Tātaihia te pūngao pūmanawa ā-papa i te pūwāhi teitei rawa.
\[ EP_{ira \, teitei rawa} = mgh \]
\[ EP_{ira \, teitei rawa} = 2 \times 9,8 \times 22,9 \]
\[ EP_{ira \, teitei rawa} \tata ki te 449,72 \, \kuputuhi{J} \]
Nō reira, ko te pūngao nekeneke o te pere i te pūwāhi teitei rawa ko 449,21 Joules, ā, ko te pūngao pūmanawa ā-papa i te pūwāhi teitei rawa ko 449,72 Joules.
Tauira Pātai 4: Pūngao Wera i roto i te Waku
Pātai: Ka panaia tētahi pouaka he 10 kg te taumaha ki runga i tētahi papa taratara mō te 5 mita me te kaha pumau o te 30 N. Ko te tauwehenga o te waku nekeneke i waenga i te pouaka me te papa he 0,2. E hia te kaha ka hurihia hei kaha wera nā te waku?
Kōrero:
Hipanga 1: Tātaihia te kaha waku.
\[ f_{waru} = \mu N \]
Dimana:
– Ko te tauwehenga o te waku (0,2) te \( \mu \)
– Ko te kaha noa te \( N \). Mō te mata papatahi, \( N = mg \)
\[ f_{waru} = 0,2 \times 10 \times 9,8 = 19,6 \, \text{N} \]
Hipanga 2: Tātaihia te mahi i mahia e te kaha waku.
\[ W = f_{waru} \times d \]
Dimana:
– Ko te tawhiti (5 m) ko te \( d \)
\[ W = 19,6 \times 5 = 98 \, \text{J} \]
Ko te pūngao i hurihia hei pūngao wera nā te waku he 98 Joules.
Whakamutunga
He ariā kaha te Ture Tiaki Pūngao e pā ana ki ngā āhuatanga ahupūngao maha. Mā te mārama ki tēnei ka taea e tātou te tātari i te whānuitanga o ngā āhuatanga ā-tinana, mai i ngā mea e hinga noa ana me te nekehanga pere ki te pūngao e puta mai ana i te waku. Mā ngā tauira i runga ake nei tātou e āwhina ki te mārama me pēhea te hurihanga o te āhua o te pūngao i te mea e mau tonu ana te rahinga. He mea nui te māramatanga hōhonu ki tēnei ariā mō te mātauranga me ngā tono mahi i roto i te hangarau me te pūtaiao.
