Range, max height, time of flight — all derived: Foundations

1.1.20 · D1 · Physics › Measurement, Vectors & Kinematics › Range, max height, time of flight — all derived

Is page par kuch bhi assumed nahi hai. Jab tak tum " $T = \frac{2 u sin θ}{g}$ " nahi padh sakte, tumhe ye jaanna chahiye ki un marks mein se har ek kya hai, woh kis picture ko represent karta hai, aur topic us ke bina kyu kaam nahi kar sakta. Hum unhe ek ek karke build karte hain, har ek pichle wale par tikka hua.

0. Woh picture jisme sab kuch rehta hai

Hum jo bhi symbol define karte hain, woh sab ek diagram mein kisi cheez ki taraf point karte hain: ek ball flat ground se launch hoti hai, arc mein udti hai, aur phir waapis landing karti hai. Yeh picture apne dimag mein rakho — hum ise ek ek karke label karte rahenge.

Figure — Range, max height, time of flight — all derived

Red arc dekho: woh ball ka path (matlab "trajectory") hai. Dashed lines woh do motions hain jo hum alag karenge. Neeche jo bhi hai, woh is picture ke kisi feature ka naam hai.

1. Do axes: horizontal aur vertical

Picture: Figure mein, $x$ ground ke saath left-to-right jaata hai, $y$ seedhi seedhi ladder ki tarah neeche se upar jaata hai.

Topic ko yeh kyu chahiye: Gravity sirf neeche kheenchti hai. Isliye upar-neeche wali motion side-to-side motion se bilkul alag behave karti hai. Duniya ko in do perpendicular directions mein split karne se hum dono ko apne simple rule se treat kar sakte hain. Yahi poora trick hai — dekho Projectile Motion — components & independence.

2. Speed aur velocity — $u$ aur $v$

Picture: Figure mein, launch point par chhota sa red arrow $u$ hai — uski length speed hai, uski tilt direction hai.

Topic ko yeh kyu chahiye: Ball ek hi push se shuru hoti hai. Uske baad sab kuch — kitni der, kitni uunchi, kitni door — is baat se decide hota hai ki woh push kitni zabardasst thi ( $u$ ) aur kis direction mein thi ( $θ$ ). Hum initial value ke liye letter $u$ aur current value ke liye $v$ use karte hain taaki hum kabhi "start" aur "abhi" ko confuse na karein.

3. Launch angle $θ$

Picture: Figure mein yeh ground aur red launch arrow ke beech wali wedge-shaped opening hai.

$θ = 0^{\circ}$ → flat phenko, ground ke saath seedha.
$θ = 9 0^{\circ}$ → seedha upar phenko.
$θ = 4 5^{\circ}$ → exactly beech mein, ek "corner" tilt.

Topic ko yeh kyu chahiye: Wahi speed $u$ tilt ke hisaab se bahut alag flights produce kar sakti hai. Flat throw ( $θ$ chhota) door jaata hai par neeche; steep throw ( $θ$ bada) uuncha jaata hai par door nahi. $θ$ woh knob hai jo ek ko doosre ke saath trade karta hai.

4. Arrow ko split karna: components $u_{x}$ aur $u_{y}$

Yeh poore topic mein sabse important move hai. Hum tilted launch arrow $u$ lete hain aur poochhhte hain: iska kitna hissa sideways point karta hai, aur kitna upar?

Picture: Figure mein red slanted arrow $u$ hai. Ground par ek seedhi line girano: ground ke saath woh shadow $u_{x}$ hai, jis height tak woh pahunchti hai woh $u_{y}$ hai. Milke $u_{x}$ (bottom) aur $u_{y}$ (side) ek right triangle banate hain jisme $u$ slanted long side hai.

Sine aur cosine kyu — aur kuch kyun nahi

Hamare paas ek right triangle hai. Uski sabse lambi side (hypotenuse, right angle ke opposite) $u$ hai. Angle $θ$ launch corner par baitha hai. Humhe "length $u$ aur angle $θ$ " ko do chhoti sides mein convert karna hai. Trigonometry ka invention exactly isi kaam ke liye hua tha.

Adjacent side (jo $θ$ ko touch karti hai) horizontal wali hai, aur opposite side (jo $θ$ ke saamne hai) vertical wali hai. Toh:

$u_{x} = u cos θ, u_{y} = u sin θ$

Topic ko yeh kyu chahiye: Ek baar $u$ , $u_{x}$ aur $u_{y}$ mein split ho jaaye, hum flat drift aur upar-neeche wali flight ko do alag 1D problems ki tarah treat kar sakte hain. Dekho Vectors — resolving into components.

5. Acceleration aur gravity — $a$ aur $g$

Picture: Figure mein, gravity ball par woh mota neeche wala arrow hai — hamesha seedha neeche point karta hai, hamesha ek hi size ka, arc ke har point par.

Sign convention: Hum upar ko positive direction choose karte hain. Chunki gravity neeche kheenchti hai, uska acceleration $a_{y} = - g$ hai (negative). Horizontally koi force nahi, isliye $a_{x} = 0$ .

Topic ko yeh kyu chahiye: Gravity hi akeli cheez hai jo path ko curve karti hai. Wahi ball ko slow karti hai, rokti hai, aur waapis girwa deti hai. $g$ ke bina ball hamesha ke liye seedhi line mein udh jaati. Deep dive: Free Fall & Acceleration due to gravity.

6. Displacement aur time — $y$ , $x$ , $t$

Picture: Red arc par koi bhi point lo. Start se uski horizontal distance $x$ hai; uski height $y$ hai; jis moment woh wahan hai woh $t$ hai.

Topic ko yeh kyu chahiye: $t$ hi glue hai. Vertical motion decide karti hai kab cheezein hoti hain (kab peak karta hai, kab land karta hai); phir hum wahi $t$ horizontal motion mein daaltein hain kahan pata karne ke liye. Yeh shared clock hi do independent motions ke beech ka poora coupling hai.

7. Woh 1D motion equations jo hum reuse karenge

Kyunki har axis ek simple 1D motion hai, hum Equations of Motion (1D kinematics) se do standard results lete hain. Vertical axis ke along ( $u_{y} = u sin θ$ aur $a_{y} = - g$ ke saath):

Do alag equations kyun? Yeh alag sawaalon ka jawaab dete hain:

Pehle mein $t$ hai — jab timing ki fikr ho tab use karo (time of flight).
Doosre mein $t$ nahi hai — jab koi velocity pata ho aur height chahiye par abhi time nahi pata. Top par, $v_{y} = 0$ pata hai par time nahi, isliye max height ke liye doosra tool perfect hai.

Sawaal ke hisaab se sahi tool choose karna ek skill hai jis par parent note baar baar rely karta hai.

8. Double-angle identity $2 sin θ cos θ = sin 2 θ$

Jab hum range calculate karte hain toh $u cos θ$ aur $u sin θ$ ka multiplication hota hai, jo combo $2 sin θ cos θ$ produce karta hai. Ek clean identity hai jo ise ek term mein fold kar deti hai:

Topic ko yeh kyu chahiye: Yeh ek nazar mein reveal karta hai ki range $θ = 4 5^{\circ}$ par sabse zyada hai (kyunki $sin 2 θ$ peak karta hai jab $2 θ = 9 0^{\circ}$ ho) aur complementary angles ek hi range share karte hain. Iske bina formula phir bhi kaam karta hai par apna secret chhupaaye rakhta hai. Full treatment: Trigonometric identities — double angle.

Yeh foundations topic ko kaise feed karte hain

Upar se neeche padho: axes, speed, angle, aur trigonometry sab milke components banate hain; components plus gravity aur shared clock 1D equations ko feed karte hain; woh equations (double-angle identity ki madad se) Range, max height, time of flight — all derived mein teen boxed results generate karte hain.

Equipment checklist

Khud ko test karo — right side dhako aur zor se jawab do.

"Horizontal" aur "vertical" mein split karne se kya fayda hota hai?

Do independent 1D motions milti hain, kyunki gravity sirf vertically act karti hai.

u

kya hai?

Launch velocity ki size — ball shuru mein kitni tez nikalti hai.

θ

kya hai aur ise kahan se measure karte hain?

Launch angle, flat ground se upar ki taraf measure kiya hua.

u_{x}

aur

u_{y}

u

aur

θ

ke terms mein likho.

u_{x} = u cos θ

u_{y} = u sin θ

Horizontal part cosine kyu use karta hai?

Angle ground se measure hota hai, isliye horizontal side

θ

ke adjacent hai, aur

cos = \frac{adjacent}{hypotenuse}

g

kya hai aur agar upar positive hai toh uske acceleration ka sign kya hoga?

Gravity ke karan acceleration (

\approx 9.8 m/s^{2}

);

a_{y} = - g

(negative, neeche ki taraf).

Flight ke dauran

a_{x}

kya hai?

Zero — koi horizontal force nahi, isliye

u_{x}

constant rehti hai.

Horizontal aur vertical motions ko kaunsi ek quantity link karti hai?

Shared time

t

v_{y}^{2} = u_{y}^{2} - 2 g y

kab use karte hain

y = u_{y} t - \frac{1}{2} g t^{2}

ki jagah?

Jab koi velocity pata ho aur height chahiye par abhi time nahi pata (jaise peak par jahan

v_{y} = 0

ho).

2 sin θ cos θ

kis cheez mein simplify hota hai?

sin 2 θ

Jab

θ = 9 0^{\circ}

ho toh

u_{x}

aur

u_{y}

kya hote hain?

u_{x} = 0

u_{y} = u

(seedha upar).