1. Motion Basics

Distance vs. Displacement

In kinematics, we must carefully distinguish between the total path traveled and the net change in position. This introduces the fundamental difference between scalar and vector quantities.

Distance (Scalar): A measure of the total length of the path moved by an object, regardless of the direction of travel. It is measured in metres ($m$).

Displacement (Vector): A measure of how far an object is from its original starting position, measured in a straight line along a specified direction. It is measured in metres ($m$).

Real-World Scenario: The Athletics Track
If an athlete completes exactly one full lap of a standard outdoor athletics track:
The distance travelled is 400 m.
The displacement is 0 m (because they returned to the exact starting position).

Speed vs. Velocity

Just as distance pairs with displacement, the rate of change of these quantities yields speed and velocity.

Speed (Scalar): A measure of how distance changes with time. Since it is dependent on distance, it is a scalar quantity.

$$\text{speed} = \dfrac{\Delta d}{\Delta t}$$

Velocity (Vector): A measure of how displacement changes with time. Since it depends on displacement, it is a vector quantity.

$$v = \dfrac{\Delta s}{\Delta t}$$

Both are measured in metres per second ($\text{m/s}$), and time is measured in seconds ($s$).

Acceleration

Acceleration is the rate at which velocity changes. Because velocity is a vector quantity, acceleration is also a vector.

Calculating Acceleration

If an object accelerates from an initial velocity $u$ to a final velocity $v$, and it takes $t$ seconds to do it, the equation is:

$$a = \dfrac{v - u}{t} \quad \text{or} \quad a = \dfrac{\Delta v}{\Delta t}$$

Acceleration is measured in metres per second squared ($\text{m/s}^2$).

Example 1

Problem: A laboratory test drone flies 60 m due East in 20 s, stops instantly, and then flies 20 m due West in 5 s. Calculate the drone's:

  1. Total distance
  2. Net displacement from its starting position
  3. Average speed
  4. Average velocity

Solution:

  • a) Total Distance: Distance is a scalar summation of the entire path length.
    Distance = 60 m + 20 m = 80 m
  • b) Net Displacement: Taking East as the positive (+) direction:
    Displacement = (+60 m) + (-20 m) = +40 m (or 40 m East)
  • c) Average Speed: Total path distance divided by total elapsed time:
    Average Speed = 80 m / (20 s + 5 s) = 80 m / 25 s = 3.2 m/s
  • d) Average Velocity: Net displacement vector divided by total elapsed time:
    Average Velocity = +40 m / 25 s = +1.6 m/s (or 1.6 m/s East)

⚠ Direction and Sign Conventions

Because these are vectors, direction is crucial. We must define which direction is positive (+ve).

  • If velocity is positive: A positive acceleration means increasing velocity (speeding up). A negative acceleration means decreasing velocity (slowing down).
  • If velocity is negative: A negative acceleration means speeding up in the negative direction. A positive acceleration means slowing down in the negative direction.

Uniform vs. Non-Uniform Acceleration

Uniform Acceleration: The acceleration is constant. The velocity changes by the same amount each unit of time.

Time ($s$) 0 1 2 3 4
Velocity ($\text{m/s}$) 0 2 4 6 8
Acceleration ($\text{m/s}^2$) - 2 2 2 2

Non-Uniform Acceleration: The acceleration is changing. The velocity changes by a different amount each unit of time.

Time ($s$) 0 1 2 3 4
Velocity ($\text{m/s}$) 0 2 6 10 18
Acceleration ($\text{m/s}^2$) - 2 4 4 8