Study Notes: Motion in a Plane

Basics of Scalars and Vectors

• Definition of Scalars: Physical quantities that possess magnitude only. A scalar is completely specified by a single numerical value and a proper unit.

  • Examples: Distance between points, mass of an object, temperature of a body, and time.

  • Rules for Combination: Scalars follow the rules of ordinary algebra. They can be added, subtracted, multiplied, and divided like ordinary numbers, provided they have the same units for addition and subtraction.

  • Example Calculations:

    • Perimeter of a rectangle with length $1.0\,m$ and breadth $0.5\,m$ is $1.0 + 0.5 + 1.0 + 0.5 = 3.0\,m$.

    • The difference between a maximum temperature of $35.6\,^{\circ}C$ and a minimum of $24.2\,^{\circ}C$ is $11.4\,^{\circ}C$.

    • A solid aluminium cube of side $10\,cm$ has a volume of $10^{-3}\,m^3$ and a density of $2.7 \times 10^3\,kg\,m^{-3}$, both of which are scalars.

• Definition of Vectors: Physical quantities that possess both magnitude and direction, and obey the triangle law or parallelogram law of addition.

  • Representation: Represented by boldface type (e.g., $\mathbf{v}$) or by an arrow over a letter (e.g., $\vec{v}$). The magnitude is called the absolute value, denoted by lightface type ($v$) or $|\mathbf{v}|$.

  • Examples: Displacement, velocity, acceleration, and force.

Position and Displacement Vectors

• Position Vector: To describe an object's position in a plane, a convenient origin $O$ is chosen. If a particle is at point $P$ at time $t$, the vector $\mathbf{OP} = \mathbf{r}$ is the position vector. Its length represents the magnitude, and its direction is the line from $O$ to $P$.

• Displacement Vector: When an object moves from point $P$ (at time $t$) to point $P'$ (at time $t'$), the vector $\mathbf{PP'}$ is the displacement vector, denoted $\Delta\mathbf{r}$.

  • Path Independence: Displacement is the straight line joining the initial and final positions. It is independent of the actual path taken. For different paths like $PABCQ$, $PDQ$, and $PBEFQ$, the displacement vector $\mathbf{PQ}$ remains identical.

  • Magnitude Relationship: The magnitude of displacement is always less than or equal to the actual path length.

Equality and Manipulation of Vectors

• Equality of Vectors: Two vectors $\mathbf{A}$ and $\mathbf{B}$ are equal if and only if they have the same magnitude and the same direction.

  • Free Vectors: In this context, vectors have no fixed locations. Displacing a vector parallel to itself leaves it unchanged.

  • Localised Vectors: In specific physical applications, the location or line of application is important, distinguishing them from free vectors.

• Multiplication by Real Numbers:

  • Multiplying $\mathbf{A}$ by a positive number $\lambda$ results in a vector with magnitude $\lambda|\mathbf{A}|$ and the same direction as $\mathbf{A}$.

  • Multiplying $\mathbf{A}$ by a negative number $-\lambda$ results in a magnitude of $\lambda|\mathbf{A}|$ and a direction opposite to $\mathbf{A}$.

  • Dimensionality: If the multiplier is a scalar with physical dimensions, the resulting vector's dimension is the product of both dimensions (e.g., velocity $\times$ time = displacement).

• Null Vector (Zero Vector): A vector with zero magnitude, denoted $\mathbf{0}$. Its direction cannot be specified.

  • Examples: Result of adding equal and opposite vectors $\mathbf{A} + (-\mathbf{A})$, or the displacement of an object that returns to its starting point.

  • Properties:

    • $\mathbf{A} + \mathbf{0} = \mathbf{A}$

    • $\lambda \mathbf{0} = \mathbf{0}$

    • $0 \mathbf{A} = \mathbf{0}$

Graphical Methods of Vector Addition and Subtraction

• Triangle Method (Head-to-Tail): To find the sum $\mathbf{A} + \mathbf{B}$, place the tail of $\mathbf{B}$ at the head of $\mathbf{A}$. The vector connecting the tail of $\mathbf{A}$ to the head of $\mathbf{B}$ is the resultant $\mathbf{R}$.

• Parallelogram Method: Bring the tails of $\mathbf{A}$ and $\mathbf{B}$ to a common origin $O$. Draw lines parallel to each vector to form a parallelogram. The diagonal from the origin represents the resultant $\mathbf{R}$.

• Laws of Addition:

  • Commutative Law: $\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}$

  • Associative Law: $(\mathbf{A} + \mathbf{B}) + \mathbf{C} = \mathbf{A} + (\mathbf{B} + \mathbf{C})$

• Subtraction: Defined as adding the negative of a vector: $\mathbf{A} - \mathbf{B} = \mathbf{A} + (-\mathbf{B})$.

Resolution of Vectors

• General Resolution: A vector $\mathbf{A}$ in a plane can be resolved into two component vectors along any two non-zero, non-collinear vectors $\mathbf{a}$ and $\mathbf{b}$ in the same plane: $\mathbf{A} = \lambda\mathbf{a} + \mu\mathbf{b}$.

• Unit Vectors: Vectors of unit magnitude used to specify direction. They have no units or dimensions.

  • $\hat{i}$, $\hat{j}$, and $\hat{k}$ denote unit vectors along the $x$, $y$, and $z$ axes respectively.

  • $|\hat{i}| = |\hat{j}| = |\hat{k}| = 1$.

  • Any vector $\mathbf{A}$ can be written as $\mathbf{A} = |\mathbf{A}|\hat{n}$, where $\hat{n}$ is the unit vector along $\mathbf{A}$.

• Rectangular Components in 2D:

  • $\mathbf{A} = A_x \hat{i} + A_y \hat{j}$

  • $A_x = A \cos(\theta)$

  • $A_y = A \sin(\theta)$

  • $A = \sqrt{A_x^2 + A_y^2}$

  • $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$

• Rectangular Components in 3D:

  • $\mathbf{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$

  • $A_x = A \cos(\alpha)$, $A_y = A \cos(\beta)$, $A_z = A \cos(\gamma)$

  • $A = \sqrt{A_x^2 + A_y^2 + A_z^2}$

Analytical Method of Vector Addition

• If $\mathbf{R} = \mathbf{A} + \mathbf{B}$, then:

  • $R_x = A_x + B_x$

  • $R_y = A_y + B_y$

  • $R_z = A_z + B_z$

• Law of Cosines: The magnitude of the resultant $R$ of two vectors $\mathbf{A}$ and $\mathbf{B}$ separated by angle $\theta$ is:

  • $R = \sqrt{A^2 + B^2 + 2AB \cos(\theta)}$

• Law of Sines:

  • $\frac{R}{\sin(\theta)} = \frac{A}{\sin(\beta)} = \frac{B}{\sin(\alpha)}$ (where $\alpha$ is the angle between $\mathbf{R}$ and $\mathbf{A}$, and $\beta$ is the angle between $\mathbf{R}$ and $\mathbf{B}$).

• Direction of Resultant:

  • $\tan(\alpha) = \frac{B \sin(\theta)}{A + B \cos(\theta)}$

Kinematics in a Plane

• Position Vector: $\mathbf{r} = x\hat{i} + y\hat{j}$.

• Displacement Vector: $\Delta\mathbf{r} = \Delta x \hat{i} + \Delta y \hat{j}$.

• Velocity:

  • Average Velocity: $\mathbf{v} = \frac{\Delta\mathbf{r}}{\Delta t} = \frac{\Delta x}{\Delta t}\hat{i} + \frac{\Delta y}{\Delta t}\hat{j}$.

  • Instantaneous Velocity: $\mathbf{v} = \frac{d\mathbf{r}}{dt} = v_x \hat{i} + v_y \hat{j}$, where $v_x = \frac{dx}{dt}$ and $v_y = \frac{dy}{dt}$.

  • Direction: The instantaneous velocity at any point is always tangential to the path at that point and in the direction of motion.

• Acceleration:

  • Average Acceleration: $\mathbf{a} = \frac{\Delta \mathbf{v}}{\Delta t} = \frac{\Delta v_x}{\Delta t}\hat{i} + \frac{\Delta v_y}{\Delta t}\hat{j}$.

  • Instantaneous Acceleration: $\mathbf{a} = \frac{d\mathbf{v}}{dt} = a_x\hat{i} + a_y\hat{j}$, where $a_x = \frac{dv_x}{dt}$ and $a_y = \frac{dv_y}{dt}$.

  • In 2D/3D motion, velocity and acceleration vectors can have any angle between $0^{\circ}$ and $180^{\circ}$.

Motion with Constant Acceleration in a Plane

• If acceleration $\mathbf{a}$ is constant and an object starts with velocity $\mathbf{v}_0$ at $t=0$:

  • Velocity: $\mathbf{v} = \mathbf{v}_0 + \mathbf{a}t$

  • Position: $\mathbf{r} = \mathbf{r}_0 + \mathbf{v}_0t + \frac{1}{2}\mathbf{a}t^2$

• Independence of Directions: Motion in a plane can be treated as two separate, simultaneous one-dimensional motions along perpendicular directions.

  • $x(t) = x_0 + v_{0x}t + \frac{1}{2}a_xt^2$

  • $y(t) = y_0 + v_{0y}t + \frac{1}{2}a_yt^2$

Projectile Motion

• Definition: An object in flight after being projected (e.g., football, baseball). It consists of two independent components:

  • Horizontal: Constant velocity ($a_x = 0$).

  • Vertical: Constant acceleration ($a_y = -g$).

• Initial Conditions:

  • Velocity components: $v_{0x} = v_0 \cos(\theta_0)$, $v_{0y} = v_0 \sin(\theta_0)$.

  • Initial position: $x_0 = 0$, $y_0 = 0$.

• Path Equation (Trajectory):

  • $y = \left(\tan(\theta_0)\right)x - \frac{g}{2(v_0 \cos(\theta_0))^2}x^2$

  • The path of a projectile is a parabola.

• Time of Maximum Height ($t_m$):

  • $t_m = \frac{v_0 \sin(\theta_0)}{g}$

• Time of Flight ($T_f$):

  • $T_f = \frac{2v_0 \sin(\theta_0)}{g} = 2t_m$

• Maximum Height ($h_m$):

  • $h_m = \frac{(v_0 \sin(\theta_0))^2}{2g}$

• Horizontal Range ($R$):

  • $R = \frac{v_0^2 \sin(2\theta_0)}{g}$

  • Range is maximum when $\theta_0 = 45^{\circ}$, giving $R_{max} = \frac{v_0^2}{g}$.

  • Ranges are equal for projection angles $(45^{\circ} + \alpha)$ and $(45^{\circ} - \alpha)$.

Uniform Circular Motion

• Definition: Motion of an object following a circular path at constant speed.

• Centripetal Acceleration ($a_c$): Even though speed is constant, direction changes continuously, leading to acceleration directed toward the center.

  • $a_c = \frac{v^2}{R}$

• Angular Speed ($\omega$):

  • $\omega = \frac{\Delta \theta}{\Delta t}$

  • Relationship to linear speed: $v = \omega R$

  • Acceleration in terms of $\omega$: $a_c = \omega^2 R$

• Period ($T$) and Frequency ($\nu$):

  • Time period $T = \frac{2\pi R}{v} = \frac{2\pi}{\omega}$

  • Frequency $\nu = \frac{1}{T} = \frac{\omega}{2\pi}$

  • $v = 2\pi R\nu$

  • $a_c = 4\pi^2 \nu^2 R$

Illustrative Examples

• Example 3.1 (Rain and Wind): Rain falls at $35\,m/s$ vertically; wind blows at $12\,m/s$ E to W.

  • Resultant speed $R = \sqrt{35^2 + 12^2} = 37\,m/s$.

  • Angle with vertical $\theta = \tan^{-1}(12/35) \approx 19^{\circ}$. Boy should hold umbrella at $19^{\circ}$ with vertical towards the East.

• Example 3.3 (Motorboat Velocity): Boat moves North at $25\,km/h$; water current is $10\,km/h$ at $60^{\circ}$ East of South.

  • Using Law of Cosines with angle $120^{\circ}$ between vectors: $R = \sqrt{25^2 + 10^2 + 2(25)(10) \cos(120^{\circ})} \approx 22\,km/h$.

  • Angle $\phi$ from North: using Law of Sines, $\phi \approx 23.4^{\circ}$.

• Example 3.4 (Particle Dynamics): Position $\mathbf{r}(t) = 3.0t\hat{i} + 2.0t^2\hat{j} + 5.0\hat{k}$.

  • Velocity $\mathbf{v}(t) = 3.0\hat{i} + 4.0t\hat{j}$.

  • Acceleration $\mathbf{a}(t) = 4.0\hat{j}\,m/s^2$ (constant along y).

  • At $t=1.0\,s$, velocity is $3.0\hat{i} + 4.0\hat{j}$, magnitude $5.0\,m/s$, and direction $53^{\circ}$ with x-axis.

• Example 3.7 (Stones thrown from cliff): Cliff height $490\,m$, horizontal throw at $15\,m/s$.

  • Time to hit: $-490 = -\frac{1}{2}(9.8)t^2 \Rightarrow t = 10\,s$.

  • Final speed: $v_x = 15$, $v_y = -98$. Total speed $v = \sqrt{15^2 + (-98)^2} \approx 99\,m/s$.

Points to Ponder

• Path length is generally greater than or equal to the magnitude of displacement.

• Average speed is greater than or equal to the magnitude of average velocity.

• Kinematic equations for uniform acceleration do not apply to uniform circular motion because the acceleration direction changes.

• The resultant acceleration in circular motion is directed towards the center only if speed is constant.

• The shape of a trajectory (e.g., straight line vs. parabola) depends on both the constant acceleration and the initial conditions (position and velocity).