projectile-motion

Chinyorwa pamusoro pemafambiro eProjectile nematambudziko emuenzaniso nemhinduro

Kumhanya kwekutanga (v)_o) uye chikamu chekukurumidza kwekutanga (v_ox uye v_oy)

Chinhu chine mafambiro eparabolic anogara aine kumhanya kwekutanga. Nekuti kufamba kweparabolic kusanganiswa kwemafambiro munzira dzakatwasuka nedzakatwasuka, kumhanya kwekutanga kunewo zvikamu zvakatwasuka nedzakatwasuka.

Kana chinhu chikafamba nenzira yekufananidzira sezviri muMifananidzo 1 ne3, ipapo kumhanya kwekutanga munzira yakatwasuka (v _ox ) uye kumhanya kwekutanga munzira yakatwasuka (v _oy ) zvinoverengerwa uchishandisa equation:

v _ox = vo _o cos θ

v _oy = vo _o sin θ

Kana chinhu chichifamba semufananidzo wechipiri, ipapo v _o = v _ox (v _oy = 0)

Kumhanya (v)_x uye v_y) uye Nzvimbo (x na y)

Kumhanya munzira dzakatwasuka uye dzakatwasuka panguva yakatarwa kunoverengwa uchishandisa equation:

v _x = v _ox = nguva dzose ( kufamba kwakafanana kwemutsara )

v _y = v _oy + gt kana v _y ² = v _oy ² + 2 gh (kufamba kwemahara)

Nzvimbo yezvinhu munzvimbo dzakatwasuka (x) uye dzakatwasuka (y) panguva yakatarwa inoverengwa uchishandisa equation:

x = v _ox t

y = v _oy t + 1⁄2 gt ²

Zvinokonzera kumhanya (v) nenzvimbo (h)

Kumhanya kunobuda panguva yakatarwa kunoverengwa uchishandisa equation:

Kutungamirirwa kwezvinhu panguva yakatarwa kunoverengwa uchishandisa equation:

Notes:

1. Chikamu chakatarwa chekufamba kweparabolic chinoonekwa sekufamba kwakafanana kwemutsara, saka vox = vx inogara iri nguva dzose

2. Chikamu chakamira chekufamba kweparabolic chinoonekwa sekufamba kwefree-fall, saka kana chinhu chikafamba parabolic, seMifananidzo 1 ne3, chikamu chakamira chespeed yechinhu pakukwirira kwepamusoro izero (v _y = 0). Kana ukakanda marble yakamira kumusoro pakukwirira kwepamusoro, chinhu chinozorora kwechinguva (v _y = 0) usati wadzokera pasi. Nokudaro, kumhanya kwechinhu chichifamba parabolic pakukwirira kwepamusoro = v _x = v _ox

3. Kana chinhu chikafamba parabolic sezvakaratidzwa paMufananidzo 2, chikamu chakamira chekufamba parabolic chinoonekwa sekufamba kwefree-fall. Kana chinhu chikafamba parabolic seMifananidzo 1 ne3 saka chikamu chakamira chekufamba parabolic chinoonekwa sekufamba kwekumusoro kwakamira).

Matambudziko emuenzaniso:

1. Bara rinopfurwa rakatwasuka nekumhanya kwekutanga kwe20 m/s. Kana pfuti iri pamusoro pevhu mamita mashanu, sarudza:

(a) nguva mumhepo

(b) kukwirira kwakanyanya

(c) daro rakatwasuka

(d) kumhanya kwebara kana rarova pasi

Solution:

Kufamba munzira yakatwasuka kunoongororwa sekufamba kwakafanana kwemutsara, ukuwo kufamba munzira yakatwasuka kunoongororwa sekufamba kwefree-fall.

Anozivikanwa:

v _ox = 20 m/s, v _oy = 0 m/s, h = 5 m, g = 9.8 m/s ²

a) Nguva iri mumhepo

Mhinduro yacho ndeyekusarudza nguva (t) mukufamba kwe free fall.

Inozivikanwa: v _oy = 0 m/s, h = 5 m, g = 9.8 m/s ²

Zvaidiwa: t

b) Kureba kwakanyanya

Kureba kwakanyanya = h = mamita mashanu.

c) Kureba kwakatwasuka (d)

Mhinduro yacho yakafanana nekuona daro riri pakufamba kwakafanana kwemutsara

Inozivikanwa: v _ox = 20 m/s, t = 1 sekondi

Aida: d

d = vt

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

d) Kumhanya kana bara rarova pasi

v _tx = v _ox = 20 m/s

v _ty = ?

Kutanga, tinoverenga kumhanya kwekupedzisira munzira yakatwasuka (vty). Mhinduro yacho yakafanana nekuona kumhanya kwekupedzisira kwekufamba kwefree-fall.

Inozivikanwa: v _oy = 0, g = 9.8 m/s ² , t = 1 s

Aida: v _ty

v _t = vo _o + gt —> vo = 0

v _t = gt

v _t = (9.8 m/s ² )(1 s)

v _t = 9.8 m/s

Kumhanya kwebara kana rarova pasi:

Kutungamira kwebara:

Nekuti v _tx iri munzira ye x-axis yakanaka (kurudyi) uye v _ty iri munzira ye y-axis isina kunaka (ichidzika pasi),

gwara rebara parinorova pasi riri -26.1 ^o kune x-axis yakanaka (ona mufananidzo uri pazasi).

2. Kambo kakapfura bara pa30 ^o kuenda kune yakatwasuka nekumhanya kwe60 m/s. Sarudza:

(a) kukwirira kwakanyanya

(b) kumhanya kwebara pakukwirira kwakanyanya

(c) nguva mumhepo

(d) daro rakatwasuka

(e) kumhanya kwebara kana rarova pasi. Ngatitii pasi rakati sandara. 🙂

Solution:

Kufamba munzira yakatwasuka kunoongororwa sekufamba kwakafanana kwemutsara, kufamba munzira yakatwasuka kunoongororwa sekufamba kwakatwasuka kunokwira kumusoro.

Inozivikanwa: v _o = 60 m/s, theta = 30 ^o.

Zvichibva padata rinozivikanwa, tinotanga taverenga zvikamu zve vertical (v _oy ) uye horizontal (v _ox ) zve startal speed (v _o ).

a) Kureba kwakanyanya (h)

Mhinduro yacho yakafanana nekuona kukwirira kwakanyanya pakufamba kwakamira kumusoro.

Anozivikanwa:

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

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

a) Kureba kwakanyanya (h)

Mhinduro yacho yakafanana nekuona kukwirira kwakanyanya pakukwira kumusoro kwakamira.

Anozivikanwa:

v _oy = 30 m/s (iyi ndiyo kumhanya kwekutanga kwebara)

v _ty = 0 m/s (Pakukwirira kwakanyanya, kumhanya kwebara repfuti = 0 m/s. Uku ndiko kumhanya kwekupedzisira.)

g = – 9.8 m/s ²

Aida: h

v _t ² = v _o ² + 2 gh

0 ² = 30 ² + 2 (-9.8) maawa

0 = 900 – 19.6 maawa

900 = maawa makumi maviri

h = 900/19.6

h = mamita 45.9

Kureba kwakanyanya kwakawanikwa nebara = mamita 45.9.

b) Kumhanya pakukwirira kwepamusoro

Pakukwirira kwepamusoro, kumhanya kuri munzira yakatwasuka = ​​0 m/s. Pakukwirira kwepamusoro, pane kumhanya kwega kuri munzira yakatwasuka. Kumhanya kuri munzira yakatwasuka pakukwirira kwepamusoro kwakaenzana nekumhanya kwekutanga munzira yakatwasuka, inova 52.2 m/s. Kutungamira kwekukurumidza munzira yakatwasuka kunogara kuripo, kureva kuti, munzira ye x-axis yakanaka (kana kufamba kwechinhu kwatsanangurwa mudhayagiramu iri pamusoro)

c) Nguva iri mumhepo

Mhinduro yacho yakafanana nekusarudza nguva (t) mukukurukura kwekufamba kwakamira kumusoro.

Anozivikanwa:

v _oy = 30 m/s (iyi ndiyo kumhanya kwekutanga kwebara munzira yakatwasuka)

g = – 9.8 m/s ²

h = 0 m (kana bara radzokera pasi, kusuduruka kwebara munzira yakatwasuka = ​​0 m)

Zvaidiwa: t

h = vot ₊ ½ gt ²

0 = (30) t + ½ (-9.8 m/s ² ) t ²

0 = (30) t – 4.9 t ²

(30) t = 4.9 t ²

30 = 4.9 t

t = 30 / 4.9

t = masekondi maviri

Nguva iri mumhepo = masekondi 6.12

d) Kureba kwakatwasuka (d)

Mhinduro yacho yakafanana nekuona daro (d) pakufamba kwakafanana kwemutsara.

Anozivikanwa:

v _ox = 52.2 m/s

t = masekondi maviri

Aida: d

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

e) Kumhanya kana bara rarova pasi

v _tx = v _ox = 52.2 m/s

v _ty = ?

Kutanga, tinoverenga kumhanya kwekupedzisira munzira yakatwasuka (v _ty ). Mhinduro ndeyekuona kumhanya kwekupedzisira pakufamba kwakatwasuka kumusoro.

Zvinodiwa: v _oy = 30 m/s, g = -9.8 m/s ² , t = 6.12 masekondi

Aida: v _ty

v _ty = v _oy + gt

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

v _ty = (30) – (60)

v _ty = -30 m/s

Chiratidzo chisina kunaka chinoratidza kuti divi rekufamba kwekupedzisira riri pasi. Cherechedza kuti kumhanya kwekutanga munzira yakatwasuka kwakaenzana nekumhanya kwekupedzisira munzira yakatwasuka.

Kumhanya kwebara kana rarova pasi:

Kwakananga bara:

Sezvo v _tx iri munzira ye x-axis yakanaka (kurudyi) uye vty iri munzira ye y-axis isina kunaka (ichidzika),

divi rekukurumidza kwebara parinorova pasi riri -30 ^o rinenge riri positive x-axis (ona mufananidzo uri pazasi).

3. Bhora rinokandwa kubva kumucheto kwechivakwa chakareba mamita makumi mashanu chine kumhanya kwekutanga kwe10 m/s. Kana bhora rikakandwa pa30o panzvimbo yakatwasuka, sarudza:

(a) nguva iyo bhora rinosvika pasi

(b) kumhanya kwebhora parinorova pasi

(c) daro rakatambanuka rinogona kusvikwa nebhora rinoyerwa kubva kumucheto kwechivako

(d) kukwirira kwakanyanya kunosvikwa nebhora

Solution:

Kutanga, tinoverenga chikamu chakamira (v _oy ) uye chikamu chakatarwa (v _ox ) chekukurumidza kwekutanga (v _o ).

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

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

a) Nguva iyo bhora rinosvika pasi

Mhinduro yacho yakafanana nekusarudza nguva (t) mukufamba kwakatwasuka kumusoro. Hukuru hwevector ine gwara riri kumusoro hunosarudzwa kuti huve hwakanaka, hukuru hwevector ine gwara riri pasi hunosarudzwa kuti huve negative. Nzvimbo yebhora painokandwa inosarudzwa senzvimbo yekunongedzera. h ihasi nekuti pamusoro pevhu iri pasi penzvimbo yekunongedzera, g ihasi nekuti gwara rekukurumidzisa kwegiravhiti riri pasi.

Anozivikanwa:

v _oy = 5 m/s, h = – 5 m, g = – 9.8 m/s ²

Zvaidiwa: t

h = vot ₊ 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

Shandisa fomura yequadratic:

Nguva iri mumhepo = nguva kubva pakakandwa bhora kuti risvike pasi = 1.64 seconds.

b) Kumhanya kwebhora kana richirova pamusoro pevhu

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

v _ty = ?

Kutanga, tinoverenga kumhanya kwekupedzisira munzira yakatwasuka (v _ty ). Mhinduro yacho yakafanana nekuona kumhanya kwekupedzisira pakufamba kwakatwasuka kumusoro.

Inozivikanwa: v _oy = 5 m/s, g = -9.8 m/s ² , t = 1.64 masekondi

Aida: v _ty

v _ty = v _oy + gt

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

v _ty = 5 – 16

v _ty = -11 m/s

Chiratidzo chisina kunaka chinoratidza kuti divi rekukurumidza kwekupedzisira riri pasi.

Kumhanya kwebara kana rarova pasi:

Kufamba kwebara = kutungamira kwekufamba kwebara kana rarova pasi:

Kumhanya kwebara kana rarova pasi:

Nekuti vtx iri munzira ye x-axis yakanaka (kurudyi) uye vty iri munzira ye y-axis isina kunaka (ichidzika),

kwakananga bara parinorova pasi i -52o zvichienderana ne x-axis yakanaka (ona mufananidzo uri pazasi).

c) Daro rakatarwa rinogona kusvikwa nebhora rinoyerwa kubva kumucheto kwechivako

Mhinduro yacho yakafanana nekuona daro rakafambwa (d) pakufamba kwakafanana kwemutsara.

Inozivikanwa: t = 1.64 masekondi, v _x = 8.7 m/s

Aida: d

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

d) Kureba kwakanyanya kunosvikwa nebhora

Inozivikanwa: v _oy = 5 m/s, v _ty = 0 m/s ( chikamu chakamira chekukurumidza pakukwirira kwepamusoro = 0 m/s), g = -9.8 m/s ².

Aida: 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 mamita

Kureba kwakanyanya kunosvikwa nebhora = 1.3 metres pamusoro pechivako = 1.3 m + 50 m = 51.3 metres pamusoro pevhu.