Tauira o ngā Pātai Kōrero mō te Whakawhanaungatanga
Ko te whanaungatanga tētahi o ngā ariā tino taketake o te ahupūngao hou, i whakaurua mai e Albert Einstein i te tīmatanga o te rautau 20. Ka matapakihia e tēnei tuhinga te ariā o te whanaungatanga me tōna pānga ki te oranga o ia rā mā roto i ngā tauira rapanga me ngā whakamārama.
Kupu Whakataki ki te Whakawhanaungatanga
E rua ngā wāhanga matua o te ariā o te whanaungatanga: ko te Ariā Motuhake o te Whakawhanaungatanga me te Ariā Whānui o te Whakawhanaungatanga. Nā te Ariā Motuhake o te Whakawhanaungatanga, i whakaputaina i te tau 1905, i huri te māramatanga o tātou ki te wāhi me te wā. I roto i tēnei ariā, i kī a Einstein ko te tere o te māramatanga te rohe tere mutunga kore e taea te hipa, ā, he rite tonu ngā ture o te ahupūngao mō ngā kaimātakitaki katoa e neke ana i te tere pumau.
I taua wā anō, ko te Ariā Whānui o te Whakawhanaungatanga, i whakaurua mai i te tau 1915, e pā ana ki te kaha ā-papatipu. E ai ki tēnei ariā, ehara te kaha ā-papatipu i te kaha tuku iho, engari he piko o te wāhi-wā i puta mai i te papatipu.
He mea tino nui kia mārama tātou ki tēnei ariā taketake i mua i te urunga atu ki ngā tauira pātai me te matapakinga mō ēnei.
Ngā Pātai Tauira me te Kōrero
Pātai 1: Te Whakawhanuitanga o te Wā
Pātai:
Ka haere tētahi kairangi moana ki tētahi whetū tawhiti i te tere o te 0,8c (ko c te tere o te mārama). Mēnā e 10 tau te roa o te haerenga ki te Ao, e hia te roa o te wā e pā ana ki te kairangi moana e ai ki tana ake karaka (te wā tika)?
Kōrero:
Ko te whakawhānui wā he āhuatanga e puta ana nā te rerekētanga o te tere whanaunga i waenga i ngā kaimātakitaki e rua. He puhoi ake te haere o te wā mō tētahi mea e neke ana e pā ana ki tētahi kaimātakitaki tūmau.
Ko te tātai mō te whakawhanuitanga o te wā ko:
\[ \Delta t' = \frac{\Delta t}{\sqrt{1 – \frac{v^2}{c^2}}}\]
Kei hea:
– Ko te \(\Delta t'\) te wā i kitea ai te nekehanga o tētahi mea.
– Ko te \(\Delta t\) te wā i kitea ai te tūnga o tētahi mea tūmau.
– Ko te \(v\) te tere o te mea e neke ana.
– Ko te tere o te mārama te \(c\).
Monohia ngā uara e mōhiotia ana ki roto i te tātai:
\[ v = 0,8c \]
\[ \Delta t = 10 \, \kuputuhi{tau} \]
\[ \Delta t' = \frac{10}{\sqrt{1 – \frac{(0,8c)^2}{c^2}}}\]
\[ \Delta t' = \frac{10}{\sqrt{1 – 0,64}}\]
\[ \Delta t' = \frac{10}{\sqrt{0,36}}\]
\[ \Delta t' = \frac{10}{0,6}\]
\[ \Delta t' \tata ki te 16.67 \, \kuputuhi{tau}\]
Nō reira, ko te wā e pā ana ki te kairangi rererangi e ai ki tana ake karaka he 16,67 tau te roa.
Pātai 2: Te Whakawhāititanga o te Roa
Pātai:
E 100 mita te roa o tētahi mea, ā, ka inehia i te wā e okioki ana. Mena kei te neke te mea i te tere o te 0,6c, he aha te roa o te mea e ai ki tētahi kaimātakitaki tūmau?
Kōrero:
Ko te whakawhāititanga o te roa he āhuatanga e poto ake ai te roa o tētahi mea e neke ana e pā ana ki tētahi kaimātakitaki i te wā e okioki ana te mea.
Ko te tātai mō te whakapoto roa ko:
\[ L = L_0 \sqrt{1 – \frac{v^2}{c^2}} \]
Kei hea:
– Ko te \(L\) te roa o te mea e neke ana.
– Ko te \(L_0\) te roa tika (te roa o te mea ina takoto kau ana).
– Ko te tere o te mea ko \(v\).
– Ko te tere o te mārama te \(c\).
Monohia ngā uara e mōhiotia ana ki roto i te tātai:
\[ L_0 = 100 \, \text{mita} \]
\[ v = 0,6c \]
\[ L = 100 \sqrt{1 – \frac{(0,6c)^2}{c^2}}\]
\[ L = 100 \sqrt{1 – 0,36}\]
\[ R = 100 \sqrt{0,64}\]
\[ L = 100 \whakareatia ki te 0,8\]
\[ L = 80 \, \text{mita}\]
Nō reira, ko te roa o te mea e neke ana, e ai ki te kaimātakitaki tūmau, he 80 mita.
Pātai 3: Papatipu Whakawhanaungatanga
Pātai:
E 2 kg te taumaha o tētahi matūriki. Mēnā kei te neke tēnei matūriki i te tere o te 0,9c, he aha te taumaha whakawhanaungatanga o te matūriki?
Kōrero:
Ko te papatipu whakawhanaungatanga ko te papatipu o tētahi mea e piki haere ana i te neke haere o te mea ki te tere o te mārama.
Ko te tātai papatipu whakawhanaungatanga ko:
\[ m = \frac{m_0}{\sqrt{1 – \frac{v^2}{c^2}}} \]
Kei hea:
– Ko te papatipu whakawhanaungatanga te \(m\).
– Ko te papatipu toenga (papatipu tika) te \(m_0\).
– Ko te tere o te mea ko \(v\).
– Ko te tere o te mārama te \(c\).
Monohia ngā uara e mōhiotia ana ki roto i te tātai:
\[ m_0 = 2 \, \text{kg} \]
\[ v = 0,9c \]
\[ m = \frac{2}{\sqrt{1 – \frac{(0,9c)^2}{c^2}}}\]
\[ m = \frac{2}{\sqrt{1 – 0,81}}\]
\[ m = \frac{2}{\sqrt{0,19}}\]
\[ m \approx \frac{2}{0,436}\]
\[ m \tata 4,59 \, \kuputuhi{kg}\]
Nō reira, ko te papatipu whakawhanaungatanga o te matūriki ina neke i te tere o te 0,9c he tata ki te 4,59 kg.
Pātai 4: E=mc^2
Pātai:
E hia te nui o te pūngao ka puta mēnā ka whakangaromia rawatia te 1 karamu o tētahi matū, e ai ki te tātai a Einstein \(E=mc^2\)?
Kōrero:
Mā te tātai rongonui a Einstein, \(E=mc^2\) ka puta he hononga tika i waenga i te papatipu (m) me te pūngao (E), ko \(c\) te tere o te mārama.
I roto i te pūnaha SI (Pūnaha Waeine o te Ao):
– Ka inehia te papatipu (m) ki ngā kirokaramu (kg).
– Ko te tere o te mārama (c) ko \(3 \times 10^8 \, \text{m/s}\).
Me tatau tātou i te pūngao e puta mai ana i te 1 karamu o tētahi matū:
– 1 karamu = 0,001 kg
\[ E = mc^2 \]
\[ E = (0,001) (3 \times 10^8)^2 \]
\[ E = (0,001) (9 \times 10^{16}) \]
\[ E = 9 \times 10^{13} \, \text{joules} \]
Nō reira, ko te pūngao e puta mai ana mēnā ka whakangaromia rawatia te 1 karamu o te matū he \(9 \times 10^{13}\) joules.
Whakamutunga
He ariā taketake, he ariā nui hoki te whanaungatanga i roto i te ahupūngao, he pānga hohonu tōna ki te whānuitanga o ngā āhuatanga ā-tinana. Mā roto i ngā tauira i kōrerohia i runga ake nei, kua kite tātou me pēhea te whakamahi i te ariā motuhake o te whanaungatanga hei mārama ki te whakawhānui wā, te whakawhāiti roa, te papatipu whanaungatanga, me te whanaungatanga i waenga i te papatipu me te pūngao.
Mā te mārama me te mahi i ēnei raruraru, ka taea e tātou te maioha ake ki te ataahua o te ariā o te whanaungatanga me ōna pānga ki te mārama ki te ao whānui.
