COORDINATE GEOMETRY & GRAPHS

WHY THIS MATTERS

Physics is fundamentally visual. The shape of a velocity-time graph tells you whether the object accelerates or decelerates. Recognising a parabola means understanding projectile motion. An isothermal process is a rectangular hyperbola. Being able to identify standard graph shapes from their equations is an essential physics skill.

§1. The Cartesian Coordinate System

Every point in a 2D plane is described by an ordered pair $(x, y)$ — the horizontal (x-axis) and vertical (y-axis) distance from the origin (0, 0).

Abscissa: The x-coordinate (horizontal position)
Ordinate: The y-coordinate (vertical position)
Origin: $(0, 0)$ — the reference point

DISTANCE FORMULA

Distance between two points $A(x_1, y_1)$ and $B(x_2, y_2)$:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Example: Find distance between $(3, 4)$ and $(0, 0)$:

$d = \sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9+16} = \sqrt{25} = \mathbf{5}$

§2. Straight Lines — The Most Common Physics Graph

Examples of lines with different slopes and intercepts

SLOPE-INTERCEPT FORM

$$y = mx + c$$

$m = \text{slope} = \tan\alpha = \frac{\Delta y}{\Delta x}$ (gradient; represents rate of change)
$c = \text{y-intercept}$ (value of $y$ when $x = 0$)

Positive Slope ($m > 0$): Line rises left to right. Uniform acceleration (v-t graph).

Negative Slope ($m < 0$): Line falls left to right. Deceleration.

Zero Slope ($m = 0$): Horizontal line. Constant velocity.

Infinite Slope: Vertical line. Instantaneous event (not physical).

⚛️ Physics Use:
• In a velocity-time graph ($v$ vs $t$): slope = acceleration.
• In a displacement-time graph ($s$ vs $t$): slope = velocity.
• In a $F$ vs $x$ graph for spring: slope = spring constant $k$.

§3. Standard Graph Shapes in Physics

1. Parabola

Parabola: $y = x^2$ (blue) and $y = -x^2$ (red) — Projectile path shape

When $y \propto x^2$ or $y = ax^2 + bx + c$, the graph is a parabola — a symmetric U-shaped (or inverted U-shaped) curve.

Opening Upward: $y = x^2$. Minimum at vertex.

Opening Downward: $y = -x^2$. Maximum at vertex. Like a projectile path!

⚛️ Projectile Trajectory:
$y= (\tan\theta_0)x - \frac{gx^2}{2u_0^2\cos^2\theta_0}$
This is of the form $y = Ax - Bx^2$, which is a downward parabola.

2. Rectangular Hyperbola

Rectangular Hyperbola: PV = constant (Boyle's Law)

When $xy = k$ (a constant), the graph is a rectangular hyperbola. As $x$ increases, $y$ decreases proportionally.

⚛️ Boyle's Law: At constant temperature, $PV = \text{const}$. A P-V graph is a rectangular hyperbola. An inverse relationship always gives this shape.

3. Circle

Equation of a circle centered at origin with radius $r$:

$$x^2 + y^2 = r^2$$

For a circle centered at $(h, k)$: $(x-h)^2 + (y-k)^2 = r^2$

4. Ellipse

Standard form: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a \neq b$. Area = $\pi ab$.

⚛️ Kepler's First Law: All planets orbit the Sun in elliptical orbits with the Sun at one focus.

5. Power Curves: $y = x^n$

Power laws: $y = x, y = x^2, y = x^3, y = \sqrt{x}$

§4. Reading Graphs in Physics — Slope & Area

KEY PRINCIPLE

In any physics graph $y$ vs $x$:

Slope of graph = $\frac{dy}{dx}$ = the derivative of $y$ with respect to $x$
Area under graph = $\int y\, dx$ = the integral of $y$ with respect to $x$

Graph	Slope gives	Area gives
$s$ vs $t$	Velocity $v$	(Not directly useful)
$v$ vs $t$	Acceleration $a$	Displacement $s$
$a$ vs $t$	Jerk	Change in velocity $\Delta v$
$F$ vs $x$	Spring constant $k$	Work done $W$
$P$ vs $V$	Isothermal compressibility	Work done by gas

Practice Questions

DRILL 1 — LINES & COORDINATES

1. Find the slope of the line joining $(2, 5)$ and $(4, 11)$.
2. Write the equation of a line with slope 3 passing through $(0, -2)$.
3. Two lines are: $y = 2x + 1$ and $y = 3x - 4$. Find their intersection point.
4. Find the distance between $(3, -2)$ and $(-1, 1)$.
5. A displacement-time graph has slope 5. What is the velocity of the object?

DRILL 2 — IDENTIFYING SHAPES

6. Identify the shape of the graph of $y = 6/x$.
7. What kind of graph is $s = 3t^2 + 2t$ (s vs t)? What is the shape?
8. A gas undergoes isothermal compression. Sketch the P-V graph and name its shape.
9. In a v-t graph for uniform acceleration, what is the interpretation of the area under the curve?
10. The trajectory of a ball is $y = x - 0.1x^2$. Identify the shape and find the maximum height ($y_{max}$).

DRILL 3 — ADVANCED ANALYSIS

11. A $F$ vs $x$ graph for a spring has slope 200 N/m. What is the work done when stretched by 0.1 m?
12. A v-t graph from 0-5s rises 0 to 20 m/s; from 5-10s is constant. Find total displacement.
13. Plot roughly the s-t graph for: Phase 1 (0-2s, 5 m/s), Phase 2 (2-4s, at rest), Phase 3 (4-6s, 3 m/s).
14. A planet has elliptical orbit ($a=6$ AU, $b=4$ AU). Find area in $\text{AU}^2$.
15. The equation $x^2 + y^2 - 6x + 8y + 20 = 0$ represents a circle. Find its center and radius.

Answer Key

Q	Answer & Method
1	$m = \frac{11-5}{4-2} = \mathbf{3}$
2	$y = 3x - 2$
3	$2x+1 = 3x-4 \implies x=5, y=11$. Point: $\mathbf{(5, 11)}$
4	$d = \sqrt{(3-(-1))^2+(-2-1)^2} = \sqrt{16+9} = \mathbf{5}$
5	Velocity = slope of s-t graph = 5 m/s
6	Rectangular Hyperbola ($xy = 6$)
7	Parabola (s is quadratic in t)
9	Area under v-t curve = displacement
10	Parabola. Max when $dy/dx = 0$: $1 - 0.2x = 0 \implies x = 5$. $y_{max} = 5 - 0.1(25) = \mathbf{2.5}$
11	$W = \frac{1}{2}kx^2 = \frac{1}{2}(200)(0.01) = \mathbf{1\text{ J}}$
12	Phase 1: Area = (1/2)(5)(20) = 50 m. Phase 2: Area = 5×20 = 100 m. Total = 150 m
14	Area = $\pi ab = \pi(6)(4) = \mathbf{24\pi \approx 75.4\text{ AU}^2}$
15	Complete the square: $(x-3)^2 + (y+4)^2 = 5$. Center: $(3, -4)$, Radius: $\sqrt{5}$