Graphs — x-t, v-t, a-t; areas and slopes meaning

1.1.18 · Physics › Measurement, Vectors & Kinematics

The core chain

Toh structure symmetric hai:

Jaana	Operation	Example
$x \to v \to a$	slope lo	slope of $x$ - $t$ gives $v$
$a \to v \to x$	area lo	area under $v$ - $t$ gives $Δ x$

Kinematic equations ko graphs se derive karna (Derivation from scratch)

Worked example Constant acceleration ⇒

$v$ aur $x$ nikalo Lo $a =$ constant. $a$ – $t$ graph ek horizontal line hai height $a$ par.

Step 1 — $v (t)$ nikalo $a$ – $t$ ke neeche area se. $0$ se $t$ tak area = rectangle = $a \cdot t$ . Yeh $Δ v = v - u$ ke barabar hai. Yeh step kyun? $a$ – $t$ ke neeche area $Δ v$ hai (yeh rate ko accumulate karta hai). $v = u + a t$

Step 2 — $Δ x$ nikalo $v$ – $t$ ke neeche area se. $v$ – $t$ graph ab ek straight line hai $u$ (at $t = 0$ ) se $v$ (at $t$ ) tak: ek trapezium. Trapezium ka area = average height × width = $\frac{u + v}{2} t$ . Yeh step kyun? $v$ – $t$ ke neeche area displacement hai. $x = \frac{u + v}{2} t$

Step 3 — combine karo. $v = u + a t$ substitute karo: $x = \frac{u + ( u + a t )}{2} t = u t + \frac{1}{2} a t^{2}$ Yeh step kyun? $v$ ko eliminate karo taaki $x$ ko known quantities $u, a, t$ mein express kar sako. $x = u t + \frac{1}{2} a t^{2}$ Geometrically: trapezium = rectangle $u t$ (base velocity) + triangle $\frac{1}{2} (a t) (t)$ (badhta hua part).

Common mistakes (Steel-man + fix)

Quick forecast-then-verify

Flashcards

What does the slope of an x–t graph represent?

Instantaneous velocity,

v = d x / d t

What does the slope of a v–t graph represent?

Acceleration,

a = d v / d t

What does the area under a v–t graph represent?

Displacement

Δ x = \int v d t

What does the area under an a–t graph represent?

Change in velocity

Δ v = \int a d t

Does area under an x–t graph have physical meaning?

Nahi (units m·s); x–t ke liye tum slope use karte ho.

On a v–t graph, what does area below the time axis mean?

Negative displacement (−direction mein motion).

How do you get distance vs displacement from a v–t graph?

Distance = |areas| ka sum; displacement = net (signed) area.

Derive

v = u + a t

from graphs.

Constant-

a

line ke neeche area =

a t = Δ v

, toh

v = u + a t

Derive

x = u t + \frac{1}{2} a t^{2}

from graphs.

Trapezium area

=

rectangle

u t

+ triangle

\frac{1}{2} a t \cdot t

Flat (horizontal) x–t line means?

Object at rest,

v = 0

A curving-up x–t graph implies what about acceleration?

Slope badh raha hai ⇒

a > 0

(+x mein speeding up).

When is an object slowing down (in terms of signs)?

Jab

v

aur

a

ke opposite signs hon.

Recall Feynman: explain to a 12-year-old

Socho ek graph ek car trip ka map hai. Position line ki steepness batati hai ki car abhi kitni fast ja rahi hai — graph par steep hill matlab zooming, flat road matlab parked. Ab speed graph: uske neeche ki space ek tank bhar'ne jaisi hai — har second tum thoda aur aage jaate ho, aur wo saare chhote chhote bits stack hokar tumhe total distance batate hain. Agar car wapas jaaye, toh wo "tank" ka part khali hona count hota hai (negative), toh aage phir wapas jaana tumhe wohi chhod sakta hai jahan se shuru hua tha, chahe tum bahut kuch move karo. Slopes neeche ladder par dekhte hain (position→speed→acceleration); areas wapas upar chadhte hain.

Connections

Instantaneous vs Average Velocity — x–t par chord ka slope vs tangent ka slope.
Equations of Uniformly Accelerated Motion — yahan directly graph areas se derive kiya.
Differentiation and Integration in Physics — slope ≡ derivative, area ≡ integral.
Free Fall and Projectile Motion — constant- $a$ graphs with sign changes.
Distance vs Displacement — signed area vs absolute area.
Vectors — Components and Signs — kyun direction (sign) v–t areas mein matter karta hai.