Learning Outcomes

Upon completing this lesson, you will be able to:

Differentiate between position, displacement, and distance.
Differentiate between speed and velocity, and define average and instantaneous velocity.
Define average and instantaneous acceleration.
State and apply the three kinematic equations for motion with constant acceleration.
Solve problems involving vertical free-fall motion.
Use differentiation and integration to relate position, velocity, and acceleration for motion with non-uniform acceleration.

Topics Covered

Fundamental Concepts of Motion (Position, Velocity, Acceleration)
Motion with Constant Acceleration
Motion with Non-uniform Acceleration

I. Fundamental Concepts of Motion

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. Rectilinear kinematics refers to motion along a straight line.

A. Position and Displacement

Position $(s)$: The position of a particle is its location along a straight-line path relative to a fixed origin. Position is a vector quantity, as it has a magnitude (distance from the origin) and a direction (positive or negative).

Displacement $(\Delta s)$: Displacement is the change in a particle's position. It is also a vector.

$$ \Delta s = s_{final} - s_{initial} $$

Distance $(d)$: Distance is the total length of the path traveled. It is a positive scalar.

B. Velocity and Speed

Velocity $(v)$: Velocity is the rate of change of position. It is a vector quantity, representing both how fast an object is moving and in what direction.

Instantaneous velocity: is the velocity at a specific moment in time and is defined as the time derivative of position.

$$ v = \frac{ds}{dt} $$

Average Velocity $(v_{avg})$: The average velocity over a time interval is the total displacement divided by the total time elapsed. It represents the constant velocity that would be needed to achieve the same displacement in the same amount of time.

$$ v_{avg} = \frac{\Delta s}{\Delta t} = \frac{s_{final} - s_{initial}}{t_{final} - t_{initial}} $$

Speed: Speed is the magnitude of velocity. It is a positive scalar that tells you how fast an object is moving without regard to its direction.

C. Acceleration

Acceleration $(a)$: Acceleration is the rate of change of velocity. It is a vector quantity. An object accelerates if its speed, direction, or both are changing.

Instantaneous acceleration: is the acceleration at a specific moment and is defined as the time derivative of velocity.

$$ a = \frac{dv}{dt} $$

Deceleration $(-a)$:

This is a common term used to describe a decrease in speed. In physics, deceleration simply means that the acceleration vector is pointing in the opposite direction to the velocity vector. For example, if a car is moving in the positive direction $(v > 0)$ and slowing down, its acceleration is negative $(a < 0)$.

We can also relate acceleration directly to position by taking the second derivative:

$$ a = \frac{d^2s}{dt^2} $$

II. Motion with Constant Acceleration

A very common type of motion occurs when the acceleration of a particle is constant. In this special case, we can derive three key equations by integrating the fundamental kinematic relations.