Ngā Pātai Ahupūngao mō te Tau 11
He ariā taketake ngā whārite i roto i te ahupūngao, he mea nui kia mārama ngā ākonga o te tau 11. Ehara i te mea ko te rahi anake te tohu a ngā whārite engari ko te ahunga anō hoki. I roto i te ahupūngao, he maha ngā rahinga e whakaatuhia ana hei whārite, pērā i te tere, te whakatere, te kaha, me te nekehanga. Ka matapakihia e tēnei tuhinga ētahi tauira o ngā raruraru whārite e kitea whānuitia ana i roto i te marautanga ahupūngao o te tau 11 me pēhea te whakaoti i aua raruraru.
Te Mārama ki ngā Wētere
He rahinga kei a ia te rahinga me te ahunga o te whārite. Kāore i rite ki te tauine, he rahinga anake tōna, ka whakaratohia e te whārite he kōrero tāpiri mō te ahunga o tētahi rahinga. Ko ngā tauira o ngā whārite i roto i te ahupūngao ko:
Tere: E whakaatu ana i te tere o te neke o tetahi mea, me te ahunga o tēnā.
– Te Kaha: E whakaatu ana i te kaha o te pana, o te tō rānei, me te ahunga e pā ai te kaha.
– Whakaterenga: E whakaatu ana i ngā panonitanga o te tere me te ahunga.
Ko te tikanga, ka whakamahia ngā reta he pere kei runga i te tuhipoka wekita, pērā i te \(\vec{A}\) me ngā reta mātotoru pērā i te A.
Ngā Mahi Wetereo Taketake
1. Tāpiritanga Wēka: Ka mahia te tāpiritanga wēka mā te tāpiri i ōna wāhanga. Mena ko \(\vec{A} = (A_x, A_y)\) me \(\vec{B} = (B_x, B_y)\), ko \(\vec{A} + \vec{B} = (A_x + B_x, A_y + B_y)\).
2. Tangohanga Wēka: Ka mahia te tangohanga wēka mā te tango i ōna wāhanga. Mena \(\vec{A} = (A_x, A_y)\) me \(\vec{B} = (B_x, B_y)\), ko \(\vec{A} – \vec{B} = (A_x – B_x, A_y – B_y)\).
3. Te Whakarea o te Tauine Tauine ki te Wetere: Ka puta mai i tēnei whakarea he wetere hou he rite te ahunga, he rerekē rānei ki te wetere taketake i runga i te tohu o te tauine tauine, engari he rerekē te rahi. Mena he tauine tauine a \(k\) ā, \(\vec{A} = (A_x, A_y)\), ko \(k\vec{A} = (kA_x, kA_y)\).
4. Te Rahi o te Wēka: Ka taea te tatau i te rahi (te rahi rānei) o te wēka \(\vec{A} = (A_x, A_y)\) mā te whakamahi i te tātai: \( |\vec{A}| = \sqrt{A_x^2 + A_y^2} \).
Ngā tauira pātai me ngā otinga
Anei ētahi tauira o ngā rapanga whārite me ā rātou otinga e kitea pinepinetia ana i ngā akoranga ahupūngao o te tau 11.
Tauira Pātai 1: Tāpiritanga Wetere
Pātai: E rua ngā whārite \(\vec{A}\) me \(\vec{B}\) he wāhanga \(\vec{A} = (3, 4)\) me \(\vec{B} = (1, 2)\). Tātaihia te tapeke \(\vec{A} + \vec{B}\).
Otinga:
\[ \vec{A} + \vec{B} = (A_x + B_x, A_y + B_y) \]
\[ \vec{A} + \vec{B} = (3 + 1, 4 + 2) \]
\[ \vec{A} + \vec{B} = (4, 6) \]
Nō reira, ko te hua o te tāpiritanga whārite \(\vec{A} + \vec{B}\) ko \((4, 6)\).
Tauira Pātai 2: Te Tangohanga Wetereo
Pātai: Ki te hoatu ngā whārite \(\vec{C} = (5, 7)\) me \(\vec{D} = (2, 3)\). Tātaihia te hua o te tangohanga \(\vec{C} – \vec{D}\).
Otinga:
\[ \vec{C} – \vec{D} = (C_x – D_x, C_y – D_y) \]
\[ \vec{C} – \vec{D} = (5 – 2, 7 – 3) \]
\[ \vec{C} – \vec{D} = (3, 4) \]
Nō reira, ko te hua o te tango i te whārite \(\vec{C} – \vec{D}\) ko \((3, 4)\).
Tauira Pātai 3: Whakarea Tauine mā te Wetere
Pātai: Mena ko te whārite \(\vec{E} = (6, 8)\) me te tauine \(k = 3\), tatauhia te hua tauine \(k\vec{E}\).
Otinga:
\[ k\vec{E} = k (E_x, E_y) \]
\[ k\vec{E} = 3 (6, 8) \]
\[ k\vec{E} = (18, 24) \]
Nō reira, ko te hua o te hua tauine \(3\vec{E}\) ko \((18, 24)\).
Tauira Pātai 4: Te Rahi o te Wētera
Pātai: Tātaihia te rahi o te whārite \(\vec{F} = (9, 12)\).
Otinga:
\[ |\vec{F}| = \sqrt{F_x^2 + F_y^2} \]
\[ |\vec{F}| = \sqrt{9^2 + 12^2} \]
\[ |\vec{F}| = \sqrt{81 + 144} \]
\[ |\vec{F}| = \sqrt{225} \]
\[ |\vec{F}| = 15 \]
Nō reira, ko te rahi o te whārite \(\vec{F}\) he 15.
Tauira Pātai 5: Te Hua o te Wētera
Pātai: E rua ngā whārite \(\vec{G}\) me \(\vec{H}\) he wāhanga \(\vec{G} = (7, 24)\) me \(\vec{H} = (-4, 3)\). Tātaihia te whārite hua mai i te tāpiritanga o ngā whārite e rua me tōna rahi.
Otinga:
Tāpiritanga wetereo:
\[ \vec{G} + \vec{H} = (G_x + H_x, G_y + H_y) \]
\[ \vec{G} + \vec{H} = (7 + (-4), 24 + 3) \]
\[ \vec{G} + \vec{H} = (3, 27) \]
Te rahi o te irahiko hua:
\[ |\vec{G} + \vec{H}| = \sqrt{(G_x + H_x)^2 + (G_y + H_y)^2} \]
\[ |\vec{G} + \vec{H}| = \sqrt{3^2 + 27^2} \]
\[ |\vec{G} + \vec{H}| = \sqrt{9 + 729} \]
\[ |\vec{G} + \vec{H}| = \sqrt{738} \]
\[ |\vec{G} + \vec{H}| \tata ki te 27.15 \]
Nō reira, ko te hua o te wetewete o \(\vec{G}\) me \(\vec{H}\) ko \((3, 27)\) me te rahi o te 27.15 pea.
Ngā Whakamahinga o ngā Wekita i roto i te Ahupūngao
He mea nui te mārama ki ngā whārite, nā te mea he maha ngā āhuatanga ā-tinana e pā ana ki ēnei. Ko ētahi tauira o ngā tono whārite i roto i te ahupūngao ko:
1. Te Kaha me te Nekehanga: I roto i te tātari kaha, ka whakamahia ngā whārite hei whakatau i te ahunga me te rahi o te kaha e pā ana ki tētahi mea.
2. Ngā Papa Hiko me ngā Papa Aukume: He mea nui ngā papa hiko me ngā papa aukume hei whakamahi i roto i te ako i te aukume hiko.
3. Te Tere me te Whakaterenga: Ko te tere me te whakaterenga he whākatere e whakamahia ana i roto i te kinematics hei whakaahua i te nekehanga o tētahi mea.
4. Te Momentum: Ko te Momentum he weke e whakaahua ana i te hua o te papatipu me te tere o tētahi mea.
Whakamutunga
Ko te mārama ki te ariā o ngā whārite me te whakamahinga o aua whārite i roto i ngā tātaitanga he pūkenga taketake mā ngā ākonga ahupūngao. Ko ngā tauira rapanga i runga ake nei e whakaatu ana i te whakamahinga o ngā mahi whārite taketake i roto i ngā momo rapanga ahupūngao. Mā te mahi tonu ki te whakaoti rapanga whārite ka āwhina i te whakapakari i te māramatanga me ngā pūkenga o ngā ākonga ki te tātari whārite, he tūāpapa nui tēnei mō ngā akoranga ahupūngao matatau.
