projectile-motion

Sengoloa se buang ka motsamao oa Projectile le mathata a sampole ka litharollo

Lebelo la pele (v)_o) le likarolo tsa lebelo la pele (v_ox le v_oy)

Ntho eo metsamao ea eona ea parabola e lulang e e-na le lebelo la pele. Hobane motsamao oa parabola ke motsoako oa metsamao ka mahlakoreng a bataletseng le a otlolohileng, lebelo la pele le boetse le na le likarolo tse bataletseng le tse otlolohileng.

Haeba ntho e tsamaya ka tsela ya papiso jwalo ka ha ho bontshitswe ho Ditshwantsho tsa 1 le 3, lebelo la pele ka lehlakoreng le otlolohileng (v _ox ) le lebelo la pele ka lehlakoreng le otlolohileng (v _oy ) di balwa ho sebediswa equation:

v _ox = vo _o cos θ

v _oy = vo _o sin θ

Haeba ntho e tsamaya ka tsela ya papiso jwalo ka setshwantsho sa 2, jwale v _o = v _ox (v _oy = 0)

Lebelo (v)_x le v_y) le Boemo (x le y)

Lebelo mahlakoreng a bataletseng le a otlolohileng ka nako e itseng le baloa ho sebelisoa equation:

v _x = v _ox = e sa fetoheng ( motsamao o ts'oanang o otlolohileng )

v _y = v _oy + gt kapa v _y ² = v _oy ² + 2 gh (motsamao o sa oeleng)

Sebaka sa lintho ka mahlakoreng a rapameng (x) le a otlolohileng (y) ka nako e itseng se baloa ho sebelisoa equation:

x = v _ox t

y = v _oy t + 1⁄2 gt ²

Sephetho sa lebelo (v) le boemo (h)

Lebelo le hlahang nakong e itseng le baloa ho sebelisoa equation:

Tsela eo lintho li tsamaeang ka eona ka nako e itseng e baloa ho sebelisoa equation:

Notes:

1. Karolo e rapameng ea motsamao oa parabola e nkoa e le motsamao o ts'oanang oa mola, kahoo vox = vx e lula e sa fetohe

2. Karolo e otlolohileng ea motsamao oa parabola e nkoa e le motsamao oa ho oa ka bolokolohi, kahoo haeba ntho e tsamaea ka parabola, joalo ka Litšoantšo tsa 1 le 3, karolo e otlolohileng ea lebelo la ntho bophahamong bo phahameng ke lefela (v _y = 0). Haeba u lahlela 'mabole holimo bophahamong bo phahameng, ntho eo e phomola motsotsoana (v _y = 0) pele e retelehela tlase. Ka hona, lebelo la ntho e tsamaisa parabola bophahamong bo phahameng = v _x = v _ox

3. Haeba ntho e tsamaya ka tsela ya parabola jwalo ka ha ho bontshitswe ho Setshwantsho sa 2, karolo e otlolohileng ya motsamao wa parabola e nkuwa e le motsamao wa ho wa ka bolokolohi. Haeba ntho e tsamaya ka tsela ya parabola jwalo ka Ditshwantsho tsa 1 le tsa 3, karolo e otlolohileng ya motsamao wa parabola e nkuwa e le motsamao o otlolohileng o yang hodimo).

Mathata a mohlala:

1. Kulo e thunngoa ka lehlakoreng le otlolohileng ka lebelo la pele la 20 m/s. Haeba sethunya se le dimithara tse 5 ho tloha fatshe, fumana:

(a) nako moeeng

(b) bophahamo bo phahameng ka ho fetisisa

(c) sebaka se bataletseng

(d) lebelo la kulo ha e otla fatshe

tharollo:

Motsamao o shebileng ka lehlakoreng le otlolohileng o sekasekwa jwalo ka motsamao o otlolohileng o tshwanang, ha motsamao o shebileng ka lehlakoreng le otlolohileng o sekasekwa jwalo ka motsamao o wang ka bolokolohi.

Tse tsejoang:

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

a) Nako moeeng

Tharollo e tšoana le ho fumana nako ea ho arohana (t) motsamaong oa ho oa ka bolokolohi.

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

Ho batloa: t

b) Bophahamo bo phahameng ka ho fetisisa

Bophahamo bo phahameng = h = limithara tse 5.

c) Sebaka se bataletseng (d)

Tharollo e tšoana le ho fumana sebaka motsamaong o ts'oanang o otlolohileng

Tse tsejoang: v _ox = 20 m/s, t = motsotsoana o le mong

Ho batloa: d

d = vt

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

d) Lebelo ha kulo e otla fatše

v _tx = v _ox = 20 m/s

v _ty = ?

Sa pele, re bala lebelo la ho qetela ka lehlakoreng le otlolohileng (vty). Tharollo e tšoana le ho fumana lebelo la ho qetela la motsamao oa ho oa ka bolokolohi.

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

Ho batloa: 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

Lebelo la kulo ha e thula fatshe:

Tsela ea kulo:

Hobane v _tx e ka lehlakoreng la x-axis e ntle (ka ho le letona) mme v _ty e ka lehlakoreng la y-axis e mpe (e theohelang tlase),

Tsela eo kulo e lebang ka yona ha e fihla fatshe ke -26.1 ^o ho ya ho x-axis e ntle (sheba setshwantsho se ka tlase).

2. Kanono e ile ea thunya kulo ka lebelo la 60 m/s ka lebelo la 30 ^o ho ea holimo.

(a) bophahamo bo phahameng ka ho fetisisa

(b) lebelo la kulo bophahamong bo phahameng ka ho fetisisa

(c) nako moeeng

(d) sebaka se bataletseng

(e) lebelo la kulo ha e otla fatshe. A re re mobu o bataletse. 🙂

tharollo:

Motsamao o shebileng ka lehlakoreng le otlolohileng o sekasekwa jwalo ka motsamao o otlolohileng o tshwanang, motsamao o shebileng ka lehlakoreng le otlolohileng o sekasekwa jwalo ka motsamao o shebileng hodimo.

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

Ho latela lintlha tse tsejoang, re qala ka ho bala likarolo tse otlolohileng (v _oy ) le tse otlolohileng (v _ox ) tsa lebelo la pele (v _ox ).

a) Bophahamo bo phahameng ka ho fetisisa (h)

Tharollo e tšoana le ho fumana bophahamo bo phahameng ka ho fetisisa motsamaong o otlolohileng o eang holimo.

Tse tsejoang:

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

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

a) Bolelele bo phahameng ka ho fetisisa (h)

Tharollo e tšoana le ho fumana bophahamo bo phahameng ka ho fetisisa motsamaong o otlolohileng o yang hodimo.

Tse tsejoang:

v _oy = 30 m/s (lena ke lebelo la pele la kulo)

v _ty = 0 m/s (Bophahamong bo phahameng ka ho fetisisa, lebelo le otlolohileng la kulo = 0 m/s. Ena ke lebelo la ho qetela.)

g = – 9.8 m/s ²

Ho batloa: h

v _t ² = v _o ² + 2 gh

0 ² = 30 ² + 2 (-9.8) h

0 = 900 – 19.6 lihora

900 = lihora tse 19.6

h = 900 / 19.6

h = limithara tse 45.9

Bophahamo bo phahameng ka ho fetisisa bo fihlelletsweng ke kulo = limithara tse 45.9.

b) Lebelo le bophahamong bo phahameng ka ho fetisisa

Bophahamong bo phahameng ka ho fetisisa, lebelo le ka lehlakoreng le otlolohileng = 0 m/s. Bophahamong bo phahameng ka ho fetisisa, ho na le lebelo le le leng feela lehlakoreng le otlolohileng. Lebelo le ka lehlakoreng le otlolohileng bophahamong bo phahameng ka ho fetisisa le lekana le lebelo la pele lehlakoreng le otlolohileng, e leng 52.2 m/s. Tsela ea lebelo lehlakoreng le otlolohileng e lula e tsitsitse, ke hore, ka lehlakoreng la x-axis e ntle (haeba motsamao oa ntho o hlalositsoe setšoantšong se kaholimo)

c) Nako moeeng

Tharollo e tšoana le ho fumana nako ea karohano (t) puisanong ea motsamao o otlolohileng o eang holimo.

Tse tsejoang:

v _oy = 30 m/s (lena ke lebelo la pele la kulo ka lehlakoreng le otlolohileng)

g = – 9.8 m/s ²

h = 0 m (ha kulo e kgutlela fatshe, ho falla ha kulo ka lehlakoreng le otlolohileng = 0 m)

Ho batloa: 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 = metsotsoana e 6.12

Nako moeeng = metsotsoana e 6.12

d) Sebaka se bataletseng (d)

Tharollo e tšoana le ho fumana sebaka (d) motsamaong o ts'oanang o otlolohileng.

Tse tsejoang:

v _ox = 52.2 m/s

t = metsotsoana e 6.12

Ho batloa: d

d = vt = (52.2 m/s)(metsotsoana e 6.12) = 319.5 m

e) Lebelo ha kulo e otla fatše

v _tx = v _ox = 52.2 m/s

v _ty = ?

Sa pele, re bala lebelo la ho qetela ka lehlakoreng le otlolohileng (v _ty ). Tharollo ke ho fumana lebelo la ho qetela motsamaong o otlolohileng o yang hodimo.

Ho batloa: v _oy = 30 m/s, g = -9.8 m/s ² , t = metsotsoana e 6.12

Ho batloa: v _ty

v _ty = v _oy + gt

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

v _ty = (30) – (60)

v _ty = -30 m/s

Letšoao le fosahetseng le bontša hore tataiso ea lebelo la ho qetela e ea tlase. Hlokomela hore lebelo la pele lehlakoreng le otlolohileng le lekana le lebelo la ho qetela lehlakoreng le otlolohileng.

Lebelo la kulo ha e otla fatše:

Tsela eo kulo e shebiloeng ka eona:

Kaha v _tx e ka lehlakoreng la x-axis e ntle (ka ho le letona) mme vty e ka lehlakoreng la y-axis e mpe (e theohelang tlase),

Tsela eo lebelo la kulo le tsamayang ka yona ha e fihla fatshe ke -30 ^o mabapi le x-axis e ntle (sheba setshwantsho se ka tlase).

3. Bolo e lahlelwa ho tloha pheletsong ya moaho o bophahamo ba dimithara tse 50 ka lebelo la pele la 10 m/s. Haeba bolo e lahlelwa ho 30o hoo e ka bang ka ho rapama, fumana:

(a) nako eo bolo e fihlang fatše ka yona

(b) lebelo la bolo ha e otla fatshe

(c) sebaka se bataletseng seo bolo e ka fihlellehang ho sona se lekanngoa ho tloha pheletsong ea moaho

(d) bophahamo bo phahameng ka ho fetisisa boo bolo e bo fihletseng

tharollo:

Sa pele, re bala karolo e otlolohileng (v _oy ) le karolo e otlolohileng (v _ox ) ea lebelo la pele (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) Nako e pakeng tsa bolo le ho fihla fatshe

Tharollo e tšoana le ho fumana nako ea karohano (t) motsamaong o otlolohileng o eang holimo. Boholo ba vekthara eo tataiso ea eona e leng holimo bo khethoa hore e be ntle, boholo ba vekthara eo tataiso ea eona e leng tlase bo khethoa hore e be mpe. Sebaka sa bolo moo e lahleloang teng se khethoa e le ntlha ea referense. h ke mpe hobane bokaholimo ba lefatše bo ka tlase ho ntlha ea referense, g ke mpe hobane tataiso ea ho potlaka ha matla a khoheli e theohela tlase.

Tse tsejoang:

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

Ho batloa: 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

Sebelisa foromo ea quadratic:

Nako moeeng = nako e pakeng tsa bolo le ho lahlelwa fatshe = metsotsoana e 1.64.

b) Lebelo la bolo ha e otla bokaholimo ba lefatše

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

v _ty = ?

Sa pele, re bala lebelo la ho qetela ka lehlakoreng le otlolohileng (v _ty ). Tharollo e tšoana le ho fumana lebelo la ho qetela motsamaong o otlolohileng o nyolohelang holimo.

Tse tsejoang: v _oy = 5 m/s, g = -9.8 m/s ² , t = 1.64 metsotsoana

Ho batloa: v _ty

v _ty = v _oy + gt

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

v _ty = 5 – 16

v _ty = -11 m/s

Letšoao le lebe le bontša hore tsela eo lebelo la ho qetela le tsamaeang ka eona e ea tlase.

Lebelo la kulo ha e otla fatše:

Tsela eo kulo e tsamayang ka yona = tsela eo kulo e tsamayang ka yona ha e fihla fatshe:

Lebelo la kulo ha e otla fatše:

Hobane vtx e ka lehlakoreng la x-axis e ntle (ka ho le letona) mme vty e ka lehlakoreng la y-axis e mpe (e theohelang tlase),

Tsela eo kulo e lebang ka yona ha e fihla fatshe ke -52o e mabapi le x-axis e ntle (sheba setshwantsho se ka tlase).

c) Sebaka se bataletseng seo bolo e ka fihlellehang ho sona se lekanngoa ho tloha pheletsong ea moaho

Tharollo e tšoana le ho fumana sebaka se tsamailoeng (d) motsamaong o ts'oanang o otlolohileng.

Tse tsejoang: t = metsotsoana e 1.64, v _x = 8.7 m/s

Ho batloa: d

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

d) Bophahamo bo phahameng ka ho fetisisa boo bolo e bo fihletseng

Tse tsejoang: v _oy = 5 m/s, v _ty = 0 m/s ( karolo e otlolohileng ea lebelo bophahamong bo phahameng ka ho fetisisa = 0 m/s), g = -9.8 m/s ².

Ho batloa: 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 limithara

Bophahamo bo phahameng ka ho fetisisa boo bolo e bo fihletseng = limithara tse 1.3 ka holimo ho moaho = 1.3 m + 50 m = limithara tse 51.3 ka holimo ho mobu.