1D Motion / Kinematics

Introduction to Vectors and Scalars

In physics, measurements can be classified as either scalars or vectors. Scalars have only magnitude (size), while vectors have both magnitude and direction. Understanding the distinction is crucial for analyzing motion and forces.

• Scalar: A quantity described by magnitude only (e.g., mass, temperature, time).

• Vector: A quantity described by both magnitude and direction (e.g., displacement, velocity, force).

• Examples:

  • "Apple weighs 5kg" – Mass (Scalar)

  • "Days are 24hr long" – Time (Scalar)

  • "It’s 60°F outside" – Temperature (Scalar)

  • "I pushed with 100N left" – Force (Vector)

  • "I walked for 10 ft" – Distance (Scalar)

  • "I walked 10 ft. east" – Displacement (Vector)

  • "I drove at 80 mph" – Speed (Scalar)

  • "I drove 80 mph west" – Velocity (Vector)

Displacement vs. Distance

Both distance and displacement measure how far an object moves, but they are fundamentally different:

• Distance (d): The total length of the path traveled, regardless of direction. Always positive and a scalar.

• Displacement (\(\Delta x\)): The straight-line change in position from the initial to the final point, including direction. Can be positive or negative and is a vector.

• In physics, positive and negative signs indicate direction.

Formulas:

• Distance:

• Displacement:

Example: If you walk 10 m east and then 6 m west, your total distance is 16 m, but your displacement is 4 m east.

Velocity and Speed

Speed and velocity both describe how fast something moves, but speed is a scalar and velocity is a vector.

• Speed (s): The rate at which distance is covered. Always positive.

• Velocity (v): The rate at which displacement changes. Can be positive or negative, indicating direction.

Formulas:

• Speed:

• Velocity:

Example: You jog 15 m in 2 s, then 9 m backwards in another 2 s. Total distance = 24 m, total displacement = 6 m. Speed = 6 m/s, velocity = 1.5 m/s (direction depends on sign).

Solving Constant and Average Velocity Problems

When velocity is constant (no acceleration), the relationship between displacement, velocity, and time is straightforward:

• Average velocity:

• To solve for any variable, rearrange the equation as needed.

Example: If a car moves from m to m in 5 s, m/s.

Acceleration

Acceleration is the rate at which velocity changes with time. It is always a vector and can result from changes in speed or direction.

• Formula:

• Acceleration can be positive (speeding up in the positive direction) or negative (slowing down or speeding up in the negative direction).

Example: If your velocity changes from 10 m/s to 30 m/s in 4 s, m/s2.

Position-Time Graphs

Position-time graphs plot an object's position (y-axis) versus time (x-axis). The slope of the graph at any point gives the velocity.

• Upward slope: Moving forward (positive velocity)

• Flat (horizontal) slope: Object is at rest (zero velocity)

• Downward slope: Moving backward (negative velocity)

• Curved graph: Indicates changing velocity (acceleration)

Velocity-Time and Acceleration-Time Graphs

Velocity-time graphs show how velocity changes over time. The slope of the velocity-time graph gives acceleration, and the area under the curve gives displacement. Similarly, the area under an acceleration-time graph gives the change in velocity.

• Displacement from velocity-time graph: Area under the curve between two times.

• Change in velocity from acceleration-time graph: Area under the curve between two times.

Formulas for area:

• Rectangle:

• Triangle:

Equations of Motion (Kinematics Equations)

For motion with constant acceleration, use the following equations (Uniformly Accelerated Motion, UAM):

• 1.

• 2.

• 3.

• 4.

To solve problems, identify the known and unknown variables, select the appropriate equation, and solve for the target variable.

Vertical Motion and Free Fall

Objects in free fall experience constant acceleration due to gravity ( m/s2 downward). The same kinematics equations apply, with if upward is positive.

• Vertical motion equations mirror horizontal ones, but use for position and for acceleration.

Catch-Up and Overtake Problems

When one object catches up to another, set their position equations equal and solve for the time or position where they meet.

• Write full position equations for both objects.

• Set and solve for the unknown.

Summary Table: Scalars vs. Vectors

Additional info: This guide covers all foundational aspects of 1D motion and kinematics, including graphical analysis, equations of motion, and problem-solving strategies for both horizontal and vertical motion. Practice problems and step-by-step solutions are essential for mastering these concepts.