Te nekehanga kaupapa

He tuhinga mō te nekehanga o te Pūrere me ngā tauira raruraru me ngā otinga

Te tere tīmatanga (v_o) me te wāhanga o te tere tīmatanga (v_ox me te v_oy)

He mea he tere tīmatanga tonu ōna nekehanga parabolic. Nā te mea he huinga nekehanga i ngā ahunga whakapae me ngā ahunga poutū te nekehanga parabolic, he wāhanga whakapae me te poutū anō hoki o te tere tīmatanga.

Mena ka neke te mea i runga i te āhua porowhita pērā i ngā Pikitia 1 me te 3, ka tatauhia te tere tīmatanga i te ahunga whakapae (v _ox ) me te tere tīmatanga i te ahunga poutū (v _oy ) mā te whakamahi i te whārite:

v _ox = v _o cos θ

v _oy = v _o sin θ

Mena kei te neke te mea i te parabolic e ai ki te pikitia 2, ko v _o = v _ox (v _oy = 0)

Tere (v_x me te v_y) me te Tūnga (x me y)

Ka tatauhia te tere i ngā ahunga whakapae me ngā ahunga poutū i tētahi wā motuhake mā te whakamahi i te whārite:

v _x = v _ox = pūmau ( nekehanga rārangi ōrite )

v _y = v _oy + gt rānei v _y ² = v _oy ² + 2 gh (nekehanga hinga-kore)

Ka tatauhia te tūranga o ngā mea i ngā ahunga whakapae (x) me te poutū (y) i tētahi wā motuhake mā te whakamahi i te whārite:

x = v _ox t

y = v _oy t + 1⁄2 gt ²

Te hua o te tere (v) me te tūnga (h)

Ka tatauhia te tere hua i tētahi wā kua tohua mā te whakamahi i te whārite:

Ka tatauhia te ahunga o ngā mea i tētahi wā motuhake mā te whakamahi i te whārite:

Notes:

1. Ko te wāhanga whakapae o te nekehanga parabolic e kiia ana ko te nekehanga raina ōrite, nō reira he pumau tonu te vox = vx

2. Ko te wāhanga poutū o te nekehanga parabolic ka tirohia ko te nekehanga hinga-kore, nō reira, ki te neke parabolic te mea, pērā i ngā Pikitia 1 me te 3, ko te wāhanga poutū o te tere o te mea i te teitei mōrahi he kore (v _y = 0). Ki te whiua e koe he māpere ki runga i te teitei mōrahi, ka okioki te mea mō tētahi wā poto (v _y = 0) i mua i te tahuritanga ki raro. Nō reira, ko te tere o te mea e neke ana i te parabolic i te teitei mōrahi = v _x = v _ox

3. Mena ka neke parabolic te mea e whakaaturia ana i te Pikitia 2, ka tirohia te wāhanga poutū o te nekehanga parabolic hei nekehanga hinga-kore. Mena ka neke parabolic te mea e rite ana ki ngā Pikitia 1 me te 3, ka tirohia te wāhanga poutū o te nekehanga parabolic hei nekehanga poutū ki runga.

Ngā tauira raruraru:

1. Ka pupuhihia he matā i te taha whakapae me te tere tīmatanga o te 20 m/s. Mena kei te 5 mita te teitei o te pū i runga ake i te whenua, tautuhia:

(a) te wā i te rangi

(b) te teitei mōrahi

(c) te tawhiti whakapae

(d) te tere o te matā ina pā ki te whenua

Rongoā:

Ka tātarihia te nekehanga whakapae pērā i te nekehanga rārangi ōrite, ko te nekehanga poutū ia ka tātarihia pērā i te nekehanga hinga-kore.

Mōhiotia:

v _ox = 20 m/s, v _oy = 0 m/s, h = 5 m, g = 9.8 m/ ^s2

a) Te wā i te rangi

Ko te otinga he whakatau i te wā (t) o te nekehanga hinga noa.

E mōhiotia ana: v _oy = 0 m/s, h = 5 m, g = 9.8 m/s ²

E hiahiatia ana: t

b) Te teitei mōrahi

Teitei mōrahi = h = 5 mita.

c) Te tawhiti whakapae (d)

He rite te otinga ki te whakatau i te tawhiti i runga i te nekehanga rārangi ōrite

E mōhiotia ana: v _ox = 20 m/s, t = 1 hēkona

E hiahiatia ana: d

d = vt

d = (20 m/s)(1 s) = 20 m

d) Te tere ina pā te matā ki te whenua

v _tx = v _ox = 20 m/s

v _ty = ?

Tuatahi, ka tatauhia e tātou te tere whakamutunga i te ahunga poutū (vty). He rite te otinga ki te whakatau i te tere whakamutunga o te nekehanga hinga kore utu.

E mōhiotia ana: v _oy = 0, g = 9.8 m/s² ^, t = 1 s

E hiahiatia ana: v _ty

vt ₌ v _o + gt —> vo = 0

vt ₌ gt

v _t = (9.8 m/s ² )(1 s)

vt ₌ 9.8 m/s

Te tere o te matā ina pā ki te whenua:

Te ahunga o te matā:

Nā te mea kei te ahunga o te tuaka-x pai a v _tx (ki te taha matau), ā , kei te ahunga o te tuaka-y kino a v _{ty (ki raro),}

Ko te ahunga o te matā ina pā ki te whenua he -26.1 ^o ki te tuaka-x pai (tirohia te pikitia i raro nei).

2. I pupuhi te pū i tētahi matā i te 30 ^o ki te whakapae me te tere o te 60 m/s. Whakatauhia:

(a) te teitei mōrahi

(b) te tere o te matā i te teitei mōrahi

(c) te wā i te rangi

(d) te tawhiti whakapae

(e) te tere o te matā ina pā ki te whenua. Mehemea he papatahi te whenua. 🙂

Rongoā:

Ka tātarihia te nekehanga whakapae pērā i te nekehanga rārangi ōrite, ka tātarihia te nekehanga poutū pērā i te nekehanga poutū ki runga.

E mōhiotia ana: v _o = 60 m/s, theta = 30 ^o.

I runga i ngā raraunga e mōhiotia ana, ka tatau tuatahi mātou i ngā wāhanga poutū (v _oy ) me ngā wāhanga whakapae (v _ox ) o te tere tīmatanga (v _o ).

a) Te teitei mōrahi (h)

He rite te otinga ki te whakatau i te teitei mōrahi i te nekehanga poutū ki runga.

Mōhiotia:

v _ox = v _o cos θ = (60)(cos 30) = (60)(0.87) = 52 m/s

v _oy = v _o hara θ = (60)(hara 30) = (60)(0.5) = 30 m/s

a) Teitei mōrahi (h)

He rite te otinga ki te whakatau i te teitei mōrahi i runga i te nekehanga poutū ki runga.

Mōhiotia:

v _oy = 30 m/s (koinei te tere tīmatanga o te matā)

v _ty = 0 m/s (I te teitei mōrahi, ko te tere poutū o te matā = 0 m/s. Koinei te tere whakamutunga.)

karamu = – 9.8 m/s ²

E hiahiatia ana: h

vt ₂ = v _o ^{2 + 2} gh

0 ² = 30 ² + 2 (-9.8) hāora

0 = 900 – 19.6 hāora

900 = 19.6 hāora

h = 900 / 19.6

h = 45.9 mita

Ko te teitei mōrahi i tutuki i te matā = 45.9 mita.

b) Te tere i te teitei mōrahi

I te teitei mōrahi, ko te tere i te ahunga poutū = 0 m/s. I te teitei mōrahi, ko te tere anake i te ahunga whakapae. Ko te tere i te ahunga whakapae i te teitei mōrahi he rite ki te tere tīmatanga i te ahunga whakapae, arā, 52.2 m/s. Ko te ahunga o te tere i te ahunga whakapae he pumau tonu, arā, i te ahunga o te tuaka-x pai (mēnā ka whakaahuahia te nekehanga o tētahi mea i roto i te hoahoa i runga ake nei).

c) Te wā i te rangi

He rite te otinga ki te whakatau i te wā (t) i roto i te matapakinga o te nekehanga poutū ki runga.

Mōhiotia:

v _oy = 30 m/s (koinei te tere tīmatanga o te matā i te ahunga poutū)

karamu = – 9.8 m/s ²

h = 0 m (ina hoki te matā ki te whenua, ko te nekehanga o te matā i te ahunga poutū = 0 m)

E hiahiatia ana: t

h = v _o t + ½ gt ²

0 = (30) t + ½ (-9.8 m/ ^s² ) ^t²

0 = (30) t – 4.9 t ²

(30) t = 4.9 t ²

30 = 4.9 tāra

t = 30 / 4.9

t = 6.12 hēkona

Te wā i te rangi = 6.12 hēkona

d) Te tawhiti whakapae (d)

He rite te otinga ki te whakatau i te tawhiti (d) i runga i te nekehanga rārangi ōrite.

Mōhiotia:

v _ox = 52.2 m/s

t = 6.12 hēkona

E hiahiatia ana: d

d = vt = (52.2 m/s)(6.12 hēkona) = 319.5 m

e) Te tere ina pā te matā ki te whenua

v _tx = v _ox = 52.2 m/s

v _ty = ?

Tuatahi, ka tatauhia e tātou te tere whakamutunga i te ahunga poutū (v _ty ). Ko te otinga he whakatau i te tere whakamutunga i runga i te nekehanga poutū ki runga.

E hiahiatia ana: v _oy = 30 m/s, g = -9.8 m/s² ^, t = 6.12 hēkona

E hiahiatia ana: v _ty

v _ty = v _oy + gt

v _ty = (30) + (-9.8)(6.12)

v _ty = (30) – (60)

v _ty = -30 m/s

Ko te tohu kino e tohu ana kei raro te ahunga o te tere whakamutunga. Kia mahara ko te tere tīmatanga i te ahunga poutū he ōrite ki te tere whakamutunga i te ahunga poutū.

Te tere o te matā ina pā ki te whenua:

Te ahunga o te matā:

I te mea kei te ahunga o te tuaka-x pai a _vtx (ki te taha matau), ā, kei te ahunga o te tuaka-y kino a vty (ki raro),

Ko te ahunga o te tere o te matā ina pā ki te whenua he -30 ^° ki te taha o te tuaka-x pai (tirohia te ahua i raro nei).

3. Ka whiua he pōro mai i te taha o tētahi whare 50 mita te teitei me te tere tīmatanga o te 10 m/s. Mēnā ka whiua te pōro ki te 30° i te taha whakapae, tautuhia:

(a) te wā e tae atu ai te pōro ki te whenua

(b) te tere o te pōro ina pā ki te whenua

(c) ko te tawhiti whakapae ka taea e te pōro te tae atu ka inehia mai i te taha o te whare

(d) te teitei mōrahi i ekehia e te pōro

Rongoā:

Tuatahi, ka tatauhia e tātou te wāhanga poutū (v _oy ) me te wāhanga whakapae (v _ox ) o te tere tīmatanga (v _o ).

v _ox = v _o cos 30 ^o = (10 m/s)(0.87) = 8.7 m/s

v _oy = v _o sin 30 ^o = (10 m/s)(0.5) = 5 m/s

a) Te wā e tae atu ai te pōro ki te whenua

He rite te otinga ki te whakatau i te wā (t) i roto i te nekehanga poutū ki runga. Ko te rahi o te ira e ahu whakarunga ana ka whiriwhiria kia pai, ko te rahi o te ira e ahu whakararo ana ka whiriwhiria kia kino. Ko te tūranga o te pōro e whiua ana ka tīpakohia hei pūwāhi tohutoro. He kino a h nā te mea kei raro iho te mata o te whenua i te pūwāhi tohutoro, he kino a g nā te mea kei raro te ahunga o te whakaterenga o te kaha ā-papa.

Mōhiotia:

v _oy = 5 m/s, h = – 5 m, g = – 9.8 m/ ^s2

E hiahiatia ana: t

h = v _o t + ½ gt ²

-5 = 5 t + ½ (-9.8) t ²

-5 = 5 t – 4.9 t ²

-4.9 t ² + 5 t + 5 = 0

Whakamahia te tātai tapawhā:

Te wā i te rangi = te wā mai i te wā i whiua ai te pōro kia tau ki te whenua = 1.64 hēkona.

b) Te tere o te pōro ina pā ki te mata o te whenua

v _tx = v _ox = v _x = 8.7 m/s

v _ty = ?

Tuatahi, ka tatauhia e tātou te tere whakamutunga i te ahunga poutū (v _ty ). He rite te otinga ki te whakatau i te tere whakamutunga i runga i te nekehanga poutū ki runga.

E mōhiotia ana: v _oy = 5 m/s, g = -9.8 m/s ² , t = 1.64 hēkona

E hiahiatia ana: v _ty

v _ty = v _oy + gt

v _ty = 5 + (-9.8)(1.64)

v _ty = 5 – 16

v _ty = -11 m/s

Ko te tohu kino e tohu ana kei raro te ahunga o te tere whakamutunga.

Te tere o te matā ina pā ki te whenua:

Ko te ahunga o te tere o te matā = te ahunga o te nekehanga o te matā ina pā ki te whenua:

Te tere o te matā ina pā ki te whenua:

Nā te mea kei te ahunga o te tuaka-x pai a vtx (ki te taha matau), ā, kei te ahunga o te tuaka-y kino a vty (ki raro),

Ko te ahunga o te matā ina pā ki te whenua he -52o ki te tuaka-x pai (tirohia te ahua i raro nei).

c) Ko te tawhiti whakapae ka taea e te pōro te tae atu ka inehia mai i te taha o te whare

He rite te otinga ki te whakatau i te tawhiti i haerea (d) i runga i te nekehanga rārangi ōrite.

E mōhiotia ana: t = 1.64 hēkona, v _x = 8.7 m/s

E hiahiatia ana: d

d = vt = (8.7 m/s)(1.64 s) = 14.3 m

d) Te teitei mōrahi i ekehia e te pōro

E mōhiotia ana: v _oy = 5 m/s, v _ty = 0 m/s ( te wāhanga poutū o te tere i te teitei mōrahi = 0 m/s), g = -9.8 m/s ².

E hiahiatia ana: h

v _ty ² = v _oy ² + 2 gh

0 m/s = (5 m/s) ² + 2(-9.8 m/s ² )(h)

0 m/s = 25 (m/s) ² + (-19.6 m/s ² )(h)

25 (m/s) ² = -19.6 m/s 2 (h)

h = 25 (m/s) ² : -19.6 m/s ² = 1.3 mita

Ko te teitei mōrahi i taea e te pōro = 1.3 mita i runga ake i te tihi o te whare = 1.3 m + 50 m = 51.3 mita i runga ake i te mata o te whenua.