Vectors, Kinematics, and Free-Fall Motion: Physics Study Notes

Scalars and Vectors

Definitions and Properties

In physics, quantities are classified as either scalars or vectors based on whether they possess direction in addition to magnitude.

• Scalar: A quantity described by a single number (its magnitude) and units. Examples: mass, temperature, speed.

• Vector: A quantity that has both magnitude and direction. Examples: displacement, velocity, acceleration, force.

• Notation: Vectors are often denoted by boldface letters or with an arrow above the symbol, e.g., v or v.

• Magnitude of a Vector: The length of the vector, denoted as $|\vec{A}|$ or simply $A$.

Example: Displacement is a vector because it describes both how far and in what direction an object has moved.

Using Vectors

Vector Addition and Subtraction

Vectors can be added or subtracted graphically (using the head-to-tail method) or algebraically (using components).

• Vector Addition: Place the tail of the second vector at the head of the first. The resultant vector is drawn from the tail of the first to the head of the last.

• Vector Subtraction: Subtracting a vector is equivalent to adding its negative (reverse direction).

• Resultant Vector: The sum of two or more vectors.

Example: If $\vec{A}$ and $\vec{B}$ are vectors, the resultant is $\vec{R} = \vec{A} + \vec{B}$.

Vector Components

Any vector in a plane can be broken down into components along the x and y axes.

• x-component: $A_x = A \cos \theta$

• y-component: $A_y = A \sin \theta$

• Reconstruction: If components are known, the magnitude and direction of the vector can be found: $A = \sqrt{A_x^2 + A_y^2}$ $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$

Example: A vector of magnitude 5 units at 30° above the x-axis has components: $A_x = 5 \cos 30^\circ = 4.33$ $A_y = 5 \sin 30^\circ = 2.5$

Combining Vectors and Vector Components

Systems of Vectors

When multiple vectors act on a point, their components can be summed to find the net effect.

• Net x-component: $R_x = A_x + B_x + C_x + \ldots$

• Net y-component: $R_y = A_y + B_y + C_y + \ldots$

• Net Resultant: $R = \sqrt{R_x^2 + R_y^2}$

Example: If three forces act on a point, sum their x and y components to find the net force.

Kinematics: Motion in One and Two Dimensions

Displacement, Velocity, and Acceleration

Kinematics is the study of motion without considering its causes. The primary quantities are displacement, velocity, and acceleration.

• Displacement ($\vec{d}$): Change in position, a vector quantity.

• Velocity ($\vec{v}$): Rate of change of displacement. $\vec{v} = \frac{d\vec{x}}{dt}$

• Acceleration ($\vec{a}$): Rate of change of velocity. $\vec{a} = \frac{d\vec{v}}{dt}$

Example: If a car moves from $x_1$ to $x_2$ in time $t$, its average velocity is $v_{avg} = \frac{x_2 - x_1}{t}$.

Free-Fall Motion

Objects Under Gravity

Free-fall describes the motion of objects under the influence of gravity alone, typically near Earth's surface.

• Acceleration due to gravity ($g$): $g \approx 9.8\,\text{m/s}^2$ downward.

• Equations of Motion for Free-Fall:

  • $y = y_0 + v_{0y} t - \frac{1}{2} g t^2$

  • $v_y = v_{0y} - g t$

  • $v_y^2 = v_{0y}^2 - 2g(y - y_0)$

• Initial Conditions:

  • If an object is dropped: $v_{0y} = 0$

  • If thrown upward: $v_{0y} > 0$

Example: A rock is thrown upward from a 100 m building with $v_{0y} = 10\,\text{m/s}$. Its position after $t$ seconds is: $y = 100 + 10t - \frac{1}{2} (9.8) t^2$

Instantaneous Acceleration and Velocity

Instantaneous values are found using derivatives or by evaluating the equations at a specific time.

• Instantaneous velocity: $v_y = v_{0y} - g t$

• Instantaneous acceleration: Always $-g$ for free-fall near Earth's surface.

Example: At $t = 2$ s, the velocity of the rock above is $v_y = 10 - 9.8 \times 2 = -9.6\,\text{m/s}$ (downward).

Projectile Motion (Additional info)

Horizontal and Vertical Components

Projectile motion involves two-dimensional motion under gravity, where the horizontal and vertical motions are independent.

• Horizontal motion: Constant velocity, $x = v_{0x} t$

• Vertical motion: Accelerated by gravity, $y = y_0 + v_{0y} t - \frac{1}{2} g t^2$

Example: A ball thrown at an angle $\theta$ with speed $v_0$ has initial components: $v_{0x} = v_0 \cos \theta$ $v_{0y} = v_0 \sin \theta$

Summary Table: Vector and Scalar Quantities

Quantity	Type	Example
Distance	Scalar	5 m
Displacement	Vector	5 m east
Speed	Scalar	10 m/s
Velocity	Vector	10 m/s north
Acceleration	Vector	9.8 m/s2 downward

Additional info: Projectile motion and vector component analysis are foundational for understanding more advanced topics in physics, such as forces and energy.