You want to know how far a body went. Which single reading gives it to you?

Option C: The area under the velocity-time graph. Only that one comes out in metres. The slope of a velocity-time graph gives acceleration, the area under an acceleration-time graph gives a change in velocity, and the area under a position-time graph is metre-seconds, which stands for nothing at all.

Graphical Analysis: JEE & NEET Physics Notes and PYQs

1Slope and area

Ball-tracking cameras in sport record where a ball is, moment by moment. From that one position–time record, software works out its speed and path. Graphs let us do the same: from one graph we can read velocity, acceleration, displacement and distance.

Everything rests on two rules. The slope of a graph is how fast the vertical quantity changes with time; the area under it is that quantity added up over time:

\text{slope} = \frac{d(y)}{dt}

The rate of change.

\text{area} = \int y\,dt

The total built up over time.

Graph	Slope gives	Area gives
x–t	velocity	—
v–t	acceleration	displacement
a–t	—	change in velocity

2Position–time graphs

On an x–t graph the slope is the velocity, the shape shows the type of motion, and the y-intercept is the starting position. At any point, the velocity is the slope of the tangent: a steeper tangent means faster, and a horizontal tangent means momentarily at rest.

v = \frac{dx}{dt} = \text{slope of the tangent}

Shape of the x–t graph	Motion
Horizontal line	at rest, $v = 0$
Straight line sloping up	constant positive velocity
Straight line sloping down	constant velocity, moving backward
Curve bending up (cup, like ∪)	velocity increasing: $a > 0$
Curve bending down (cap, like ∩)	velocity decreasing: $a < 0$

3Velocity–time graphs

On a v–t graph	It gives
the height	the velocity at that moment
the slope	the acceleration
the area under it	the displacement
a horizontal line	constant velocity, $a = 0$
a line sloping down (above the axis)	slowing down, $a < 0$

Why is the area the displacement? In a tiny time $dt$ the object moves $v\,dt$ , a thin strip under the graph. Adding all the strips gives the area:

\text{displacement} = \int_{t_1}^{t_2} v\,dt

4Acceleration–time graphs and the graph ladder

The area under an a–t graph is the change in velocity. To get the velocity itself, add the starting velocity:

\Delta v = \int_{t_1}^{t_2} a\,dt

v_{\text{final}} = u + \text{area}

The graph ladder. Taking slopes moves down: x–t → v–t → a–t. Taking areas moves up: a–t → v–t → x–t. To draw one graph from another, go point by point: for the graph below, plot the slope at each moment; for the graph above, plot the area built up so far.

Motion	x–t	v–t	a–t
At rest	horizontal line	on the t-axis	on the t-axis
Constant velocity	slanted straight line	horizontal line	on the t-axis
Constant acceleration from rest	parabola starting flat	slanted line through the origin	horizontal line
Ball thrown straight up	cap-shaped parabola	straight line crossing the t-axis at the top	horizontal line at $-g$

5Impossible graphs

Some graphs can never describe real one-dimensional motion:

A vertical segment on an x–t graph: the object would be in two places at once, or jump in zero time.
A double-valued graph (a loop): two positions or two velocities at the same instant.
A graph going back in time: time only increases.
A sharp corner on an x–t graph: the velocity changes instantly, which needs an infinite acceleration. Problems use it only as an idealisation.

6A journey in one sketch

The v–t graph is the VIP: it is the only one where both the slope (acceleration) and the area (displacement) carry physics. For a journey in several parts, sketch the v–t graph first.

Summary

Key ideas

The slope of a graph is a rate of change; the area under it is a total over time.
On an x–t graph the slope of the tangent is the velocity; a horizontal tangent means momentarily at rest.
A straight x–t line means constant velocity; a steeper line means faster.
Curvature of x–t shows acceleration: cup (∪) means a > 0, cap (∩) means a < 0, straight means a = 0.
A curved x–t graph does not mean a curved path.
On a v–t graph the height is v, the slope is a, and the area is the displacement.
Area below the time axis is negative displacement; the distance adds every area as positive.
The area under an a–t graph is the change in velocity; add u to get v.
Slopes move down the ladder x → v → a; areas move up a → v → x.
A graph is impossible if a vertical line cuts it twice, it jumps vertically, runs back in time, or (for x–t) has a sharp corner.
For multi-part journeys, sketch the v–t graph and add up simple areas.

Every equation

Slope: $\text{slope} = \frac{d(y)}{dt}$
Area: $\text{area} = \int y\,dt$
Velocity from x–t: $v = \frac{dx}{dt}$
Acceleration from v–t: $a = \frac{dv}{dt}$
Displacement from v–t: $s = \int_{t_1}^{t_2} v\,dt$
Change in velocity from a–t: $\Delta v = \int_{t_1}^{t_2} a\,dt$
Final velocity: $v = u + \text{area under } a\text{–}t$
Trapezium area (straight v–t): $s = \tfrac{1}{2}(u + v)\,t$

Graphical Analysis: notes and previous year questions

Graphical Analysis in short