Motion in a Plane — Revision Notes

⚡ 30-Second Revision

Position Vector — $\vec{r} = x\hat{i} + y\hat{j}$
Displacement — $\Delta\vec{r} = \vec{r}_f - \vec{r}_i$
Velocity — $\vec{v} = d\vec{r}/dt = v_x\hat{i} + v_y\hat{j}$
Acceleration — $\vec{a} = d\vec{v}/dt = a_x\hat{i} + a_y\hat{j}$
Equations of Motion (constant $\vec{a}$)

- $\vec{v} = \vec{u} + \vec{a}t$ - $\vec{s} = \vec{u}t + \frac{1}{2}\vec{a}t^2$

Projectile Motion (from ground, $u$ at $\theta$)

- $u_x = u\cos\theta$, $u_y = u\sin\theta$ - $a_x = 0$, $a_y = -g$ - Time of Flight: $T = \frac{2u\sin\theta}{g}$ - Max Height: $H = \frac{u^2\sin^2\theta}{2g}$ - Horizontal Range: $R = \frac{u^2\sin(2\theta)}{g}$ - Trajectory: $y = x\tan\theta - \frac{gx^2}{2u^2\cos^2\theta}$

Uniform Circular Motion (UCM)

- Linear-Angular relation: $v = r\omega$ - Centripetal Acceleration: $a_c = \frac{v^2}{r} = r\omega^2$

Relative Velocity — $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$

2-Minute Revision

Motion in a plane extends 1D kinematics to two dimensions, requiring vectors for position, displacement, velocity, and acceleration. The core principle is the independence of perpendicular motions: horizontal and vertical components are analyzed separately, linked only by time.

For constant acceleration, 1D kinematic equations apply to each component. Projectile motion, a key example, involves constant horizontal velocity ($u\cos\theta$) and vertical motion under gravity ($u\sin\theta - gt$).

Key formulas for time of flight ($T = 2u\sin\theta/g$), max height ($H = u^2\sin^2\theta/2g$), and range ($R = u^2\sin(2\theta)/g$) are essential. Uniform Circular Motion (UCM) features constant speed but changing velocity, leading to centripetal acceleration ($a_c = v^2/r$) directed towards the center.

Relative velocity in 2D uses vector subtraction ($\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$) for scenarios like river crossings or rain falling on a moving person. Always resolve vectors into components and draw diagrams.

5-Minute Revision

Motion in a plane is the study of objects moving in two dimensions, typically described using x and y coordinates. All kinematic quantities (position $\vec{r}$, displacement $\Delta\vec{r}$, velocity $\vec{v}$, acceleration $\vec{a}$) are vectors. The position vector is $\vec{r} = x\hat{i} + y\hat{j}$. Velocity is $\vec{v} = d\vec{r}/dt = v_x\hat{i} + v_y\hat{j}$, and acceleration is $\vec{a} = d\vec{v}/dt = a_x\hat{i} + a_y\hat{j}$.

The most important concept is the independence of perpendicular motions. This means motion along the x-axis does not affect motion along the y-axis, allowing us to apply 1D kinematic equations separately to each component. Time is the common parameter.

Projectile Motion: An object launched into the air, moving under gravity. Neglecting air resistance, horizontal acceleration $a_x = 0$, so $v_x = u_x = u\cos\theta$ is constant. Vertical acceleration $a_y = -g$. Key formulas:

Time of Flight ($T$): $T = \frac{2u\sin\theta}{g}$
Maximum Height ($H$): $H = \frac{u^2\sin^2\theta}{2g}$
Horizontal Range ($R$): $R = \frac{u^2\sin(2\theta)}{g}$. Max range at $\theta = 45^circ$.
Equation of Trajectory: $y = x\tan\theta - \frac{gx^2}{2u^2\cos^2\theta}$

Uniform Circular Motion (UCM): Object moves in a circle at constant speed $v$. Velocity direction continuously changes, so there is acceleration. This is centripetal acceleration ($a_c$), directed towards the center. Its magnitude is $a_c = \frac{v^2}{r} = r\omega^2$, where $\omega = v/r$ is angular velocity. The force causing this is centripetal force ($F_c = ma_c$).

Relative Velocity: The velocity of an object A as observed from a moving object B is $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$. This is crucial for:

River-Boat Problems — $\vec{v}_{boat, ground} = \vec{v}_{boat, water} + \vec{v}_{water, ground}$. For shortest time, boat heads perpendicular to river. For shortest path, boat heads upstream such that resultant velocity is perpendicular to river.
Rain-Man Problems — $\vec{v}_{rain, man} = \vec{v}_{rain, ground} - \vec{v}_{man, ground}$.

Example: A ball is thrown horizontally from a $45,\text{m}$ high cliff with a speed of $15,\text{m/s}$. Find the time to hit the ground ($g=10,\text{m/s}^2$).

Vertical motion: $u_y = 0$, $s_y = -45,\text{m}$, $a_y = -10,\text{m/s}^2$. Using $s_y = u_yt + \frac{1}{2}a_yt^2$: $-45 = 0 \cdot t + \frac{1}{2}(-10)t^2 \implies -45 = -5t^2 \implies t^2 = 9 \implies t = 3,\text{s}$.
Horizontal distance: $s_x = u_xt = 15,\text{m/s} \times 3,\text{s} = 45,\text{m}$.

Always draw vector diagrams, resolve components, and apply the correct kinematic equations. Be careful with signs for direction.

Prelims Revision Notes

1
Vector Basics — Vectors have magnitude and direction. Scalars only magnitude. Position $\vec{r} = x\hat{i} + y\hat{j}$. Displacement $\Delta\vec{r} = \vec{r}_f - \vec{r}_i$. Velocity $\vec{v} = d\vec{r}/dt$. Acceleration $\vec{a} = d\vec{v}/dt$. Vector addition/subtraction is component-wise. Magnitude of $\vec{A} = A_x\hat{i} + A_y\hat{j}$ is $|vec{A}| = \sqrt{A_x^2 + A_y^2}$. Direction $\tan\theta = A_y/A_x$.
2
Independence of Perpendicular Motions — The key concept. Horizontal and vertical motions are independent. Time is the link. Apply 1D equations separately to x and y components.
3
Equations of Motion (Constant Acceleration)

* $v_x = u_x + a_xt$, $v_y = u_y + a_yt$ * $x = u_xt + \frac{1}{2}a_xt^2$, $y = u_yt + \frac{1}{2}a_yt^2$ * $v_x^2 = u_x^2 + 2a_xx$, $v_y^2 = u_y^2 + 2a_yy$

1
Projectile Motion (Ground to Ground)

* Initial velocity $u$ at angle $\theta$ with horizontal. * Horizontal component: $u_x = u\cos\theta$, $a_x = 0$, $v_x = u_x$ (constant). * Vertical component: $u_y = u\sin\theta$, $a_y = -g$. * Time of Flight ($T$): $T = \frac{2u\sin\theta}{g}$.

* Maximum Height ($H$): $H = \frac{u^2\sin^2\theta}{2g}$. (At max height, $v_y = 0$). * Horizontal Range ($R$): $R = \frac{u^2\sin(2\theta)}{g}$. Max range at $\theta = 45^circ$ ($R_{max} = u^2/g$). Ranges are equal for complementary angles ($\theta$ and $90^circ - \theta$).

* Velocity at any time $t$: $\vec{v}(t) = (u\cos\theta)\hat{i} + (u\sin\theta - gt)\hat{j}$.

1
Uniform Circular Motion (UCM)

* Speed $v$ is constant, but velocity $\vec{v}$ changes direction, hence acceleration exists. * Angular velocity $\omega = \frac{Delta\theta}{Delta t}$. Relation: $v = r\omega$. * Centripetal Acceleration ($a_c$): Always directed towards the center. $a_c = \frac{v^2}{r} = r\omega^2$. It is perpendicular to $\vec{v}$. * Centripetal Force ($F_c$): $F_c = ma_c = \frac{mv^2}{r} = mr\omega^2$. This is the net force causing the circular motion.

1
Relative Velocity in 2D

* General formula: $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$. * River-Boat Problems: $\vec{v}_{boat, ground} = \vec{v}_{boat, water} + \vec{v}_{water, ground}$. * Shortest Time: Boat heads perpendicular to river flow.

Time $T = \frac{\text{Width}}{v_{boat, water}}$. Drift $x = v_{water, ground} \cdot T$. * Shortest Path: Resultant velocity is perpendicular to river flow. Boat heads upstream at angle $\theta$ such that $v_{boat, water}\sin\theta = v_{water, ground}$.

Time $T = \frac{\text{Width}}{\sqrt{v_{boat, water}^2 - v_{water, ground}^2}}$. * Rain-Man Problems: $\vec{v}_{rain, man} = \vec{v}_{rain, ground} - \vec{v}_{man, ground}$. Draw vector diagrams to find magnitude and direction of apparent rain velocity.

Vyyuha Quick Recall

For Projectile Motion formulas, remember 'T-H-R': Time of flight: Two Up Side Gravity ($2u\sin\theta/g$) Height: Half Up Side Square Gravity ($u^2\sin^2\theta/2g$) Range: Up Side Twice Gravity ($u^2\sin(2\theta)/g$)

For Relative Velocity, think 'A relative to B is A minus B': $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$. Always remember the 'minus' for relative velocity, and then use vector addition for resultant velocities like $\vec{v}_{BG} = \vec{v}_{BW} + \vec{v}_{WG}$.