Prerequisites
Before studying this topic, make sure you understand:
- Motion in a Straight Line - 1D motion concepts
- Basic vector operations (addition, subtraction, components)
Introduction
In 2D motion, objects move in a plane (like a ball thrown at an angle, or a car turning on a road). We need vectors to describe such motion completely.
Position Vector
The position of a point P in a plane is described by the position vector:
$$\vec{r} = x\hat{i} + y\hat{j}$$where:
- $x$ = x-coordinate (horizontal position)
- $y$ = y-coordinate (vertical position)
- $\hat{i}$, $\hat{j}$ = unit vectors along x and y axes
Magnitude of Position Vector
$$|\vec{r}| = r = \sqrt{x^2 + y^2}$$Direction
$$\theta = \tan^{-1}\left(\frac{y}{x}\right)$$Displacement in 2D
When an object moves from position $\vec{r}_1$ to $\vec{r}_2$:
$$\vec{s} = \vec{r}_2 - \vec{r}_1 = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j}$$ $$|\vec{s}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$Velocity in 2D
Average Velocity
$$\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t} = \frac{\vec{r}_2 - \vec{r}_1}{t_2 - t_1}$$Instantaneous Velocity
$$\vec{v} = \frac{d\vec{r}}{dt} = \frac{dx}{dt}\hat{i} + \frac{dy}{dt}\hat{j} = v_x\hat{i} + v_y\hat{j}$$Speed (Magnitude of Velocity)
$$|\vec{v}| = v = \sqrt{v_x^2 + v_y^2}$$Direction of Velocity
$$\theta = \tan^{-1}\left(\frac{v_y}{v_x}\right)$$Interactive Demo: Visualize Projectile Motion
See how 2D motion combines independent horizontal and vertical components.
Acceleration in 2D
Average Acceleration
$$\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v}_2 - \vec{v}_1}{t_2 - t_1}$$Instantaneous Acceleration
$$\vec{a} = \frac{d\vec{v}}{dt} = \frac{dv_x}{dt}\hat{i} + \frac{dv_y}{dt}\hat{j} = a_x\hat{i} + a_y\hat{j}$$Magnitude of Acceleration
$$|\vec{a}| = a = \sqrt{a_x^2 + a_y^2}$$Key Principle: Independence of Components
In 2D motion, the x and y components are independent of each other:
- Motion along x doesn’t affect motion along y
- Motion along y doesn’t affect motion along x
This allows us to analyze complex 2D problems as two separate 1D problems!
Equations for 2D Motion (constant acceleration)
Along x-axis:
- $v_x = u_x + a_x t$
- $x = u_x t + \frac{1}{2}a_x t^2$
- $v_x^2 = u_x^2 + 2a_x x$
Along y-axis:
- $v_y = u_y + a_y t$
- $y = u_y t + \frac{1}{2}a_y t^2$
- $v_y^2 = u_y^2 + 2a_y y$
Summary Table
| Quantity | Vector Form | Components |
|---|---|---|
| Position | $\vec{r} = x\hat{i} + y\hat{j}$ | $r = \sqrt{x^2 + y^2}$ |
| Velocity | $\vec{v} = v_x\hat{i} + v_y\hat{j}$ | $v = \sqrt{v_x^2 + v_y^2}$ |
| Acceleration | $\vec{a} = a_x\hat{i} + a_y\hat{j}$ | $a = \sqrt{a_x^2 + a_y^2}$ |
Practice Problems
A particle has position $\vec{r} = (3t^2 - 2)\hat{i} + (4t)\hat{j}$ m. Find velocity and acceleration at t = 1s.
Solution: $\vec{v} = \frac{d\vec{r}}{dt} = 6t\hat{i} + 4\hat{j}$ At t = 1s: $\vec{v} = 6\hat{i} + 4\hat{j}$ m/s Speed: $|\vec{v}| = \sqrt{36 + 16} = \sqrt{52} ā $ 7.2 m/s
$\vec{a} = \frac{d\vec{v}}{dt} = 6\hat{i}$ m/sĀ² (constant)
An object moves from point A(2, 3) to B(5, 7) in 2 seconds. Find average velocity.
Solution: $\vec{s} = (5-2)\hat{i} + (7-3)\hat{j} = 3\hat{i} + 4\hat{j}$ m
$\vec{v}_{avg} = \frac{\vec{s}}{t} = \frac{3\hat{i} + 4\hat{j}}{2} = 1.5\hat{i} + 2\hat{j}$ m/s
Speed: $|\vec{v}_{avg}| = \sqrt{1.5^2 + 2^2} = $ 2.5 m/s
Applications
Understanding 2D motion is essential for:
- Projectile Motion - motion under gravity
- Circular Motion - motion along a curved path
- Relative Velocity - motion in different reference frames