Kinematics and Motion: Scalars, Vectors, and Displacement Study Notes

Scalars and Vectors

Definitions and Classification

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

• Scalar: A quantity with only magnitude (size), no direction. Examples: speed, mass, temperature.

• Vector: A quantity with both magnitude and direction. Examples: velocity, displacement, force.

Examples of Scalars and Vectors:

• 25 m/s – Vector (if direction is specified, otherwise scalar)

• The distance between your home and the school – Scalar

• Pacing back and forth – Vector (if displacement is considered)

• The speed at which the moon orbits the earth – Scalar

• A bag pulling up a wall at 3.2 m/min – Vector

• 4.7 m/s2 @ 42° N – Vector

• The mass of a falling object – Scalar

• 9.8 m/s2 – Vector (acceleration due to gravity)

Key Points:

• Vectors have both magnitude and direction; scalars have only magnitude.

• Vector addition requires consideration of both magnitude and direction.

• Speed is a scalar; velocity is a vector.

Vector Addition

Combining Displacements

When multiple displacements occur in different directions, vector addition is used to determine the overall displacement. This often involves using the Pythagorean Theorem for perpendicular vectors.

Student A	Student B
6 m, North 12 m, South 4 m, West 2 m, South 14 m, South	8 m, North 10 m, West 12 m, South 15 m, South 4 m, West
Total Distance = Displacement =	Total Distance = Displacement =

Example: If a student walks 6 m North, then 12 m South, the total distance is 18 m, but the displacement is 6 m South (net movement).

Formula:

• For perpendicular vectors: $d = \sqrt{(x^2 + y^2)}$

Speed, Velocity, and Acceleration

Definitions and Calculations

Speed is the rate at which an object covers distance (scalar), while velocity is speed with a specified direction (vector). Acceleration is the rate of change of velocity.

• Speed: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

• Velocity: $\vec{v} = \frac{\text{Displacement}}{\text{Time}}$

• Acceleration: $a = \frac{\Delta v}{\Delta t}$

Example: A cheetah runs at 53 mph. To convert to m/s and km/h:

• 1 mph ≈ 0.44704 m/s

• 53 mph × 0.44704 = 23.7 m/s

• 53 mph × 1.60934 = 85.3 km/h

Distance vs. Displacement

Comparisons and Situational Examples

Distance is the total path length traveled (scalar), while displacement is the straight-line change in position from start to end (vector).

• If you walk in a circle and return to your starting point, your displacement is zero, but your distance is the circumference of the circle.

• If you walk in a straight line, distance and displacement are equal.

Additional info: Displacement can be zero even if distance is non-zero, such as in round trips.

Graphical Analysis of Motion

Velocity-Time and Position-Time Graphs

Graphs are used to analyze motion, showing how velocity or position changes over time.

• Velocity-Time Graph: The area under the curve represents displacement.

• Position-Time Graph: The slope of the graph gives velocity.

Example: For a velocity-time graph, calculate displacement by finding the area under the curve for each segment.

Projectile Motion and Free Fall

Equations and Applications

Objects in free fall or projectile motion are subject to constant acceleration due to gravity ($g = 9.8\,\text{m/s}^2$).

• Displacement: $y = v_0 t + \frac{1}{2} a t^2$

• Final velocity: $v = v_0 + a t$

• Time to fall: $t = \sqrt{\frac{2h}{g}}$

Example: A jack-o-lantern dropped from 12.5 m:

• Initial velocity $v_0 = 0$

• Time to fall: $t = \sqrt{\frac{2 \times 12.5}{9.8}} \approx 1.6\,\text{s}$

• Final velocity: $v = 0 + 9.8 \times 1.6 \approx 15.7\,\text{m/s}$

Problem Solving in Kinematics

Worked Examples

Solving kinematics problems involves identifying knowns and unknowns, selecting appropriate equations, and applying vector addition when necessary.

• Convert units as needed (e.g., mph to m/s).

• Use vector diagrams for multi-directional motion.

• Apply the Pythagorean Theorem for resultant displacement.

Example: If George walks 750 m East and 900 m South, his resultant displacement is:

• $d = \sqrt{750^2 + 900^2} = \sqrt{562500 + 810000} = \sqrt{1372500} \approx 1171.5\,\text{m}$

Summary Table: Scalars vs. Vectors

Quantity	Scalar	Vector
Distance	✔
Displacement		✔
Speed	✔
Velocity		✔
Acceleration		✔
Mass	✔

Additional info: This table summarizes the classification of common physical quantities as scalars or vectors.