A1-1. 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 represented by $l$ and 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 represented by $\Delta x$ and is measured in metres (m).

Average Speed and Average Velocity

Just as distance pairs with displacement, the rate of change of these quantities over a specific time interval yields average speed and average velocity.

Average Speed (Scalar): The ratio of the total path length ($l$) to the time interval ($\Delta t$). Since it is dependent on distance, it is a scalar quantity.

$$\overline{v_s} = \dfrac{l}{\Delta t}$$

Average Velocity (Vector): The ratio of the displacement ($\Delta x$) to the time interval ($\Delta t$). Since it depends on displacement, it is a vector quantity.

$$\overline{v} = \dfrac{x - x_0}{t - t_0} = \dfrac{\Delta x}{\Delta t}$$

Both are measured in metres per second (m/s), and time is measured in seconds (s).

Note (Uniform Acceleration Only): When an object moves with constant (uniform) acceleration, the average velocity or average speed can also be computed using the arithmetic mean of the initial and final values:

$$\overline{v} = \dfrac{v_i + v_f}{2}$$

where $v_i$ is the initial speed/velocity and $v_f$ is the final speed/velocity.

Example 1

Problem: Alice runs 60 m in a straight line in 10 s. She stops, and then spends 50 s walking slowly back to her starting point. Taking her running direction as positive, calculate:

  1. Her average velocity while running
  2. Her average velocity while walking back
  3. Her average velocity for the entire round trip
  4. Her average speed for the entire round trip

Solution:

  • a) Average velocity (running):
    $$\overline{v} = \dfrac{\Delta x}{\Delta t} = \dfrac{60}{10} = 6.0\text{ m/s}$$ Average velocity = 6.0 m/s (forward/positive direction)
  • b) Average velocity (walking back):
    $$\overline{v} = \dfrac{\Delta x}{\Delta t} = \dfrac{-60}{50} = -1.2\text{ m/s}$$ Average velocity = -1.2 m/s (backward/negative direction)
  • c) Average velocity (round trip):
    The net displacement for returning to the start is zero.
    $$\overline{v} = \dfrac{\Delta x}{\Delta t} = \dfrac{0}{60} = 0\text{ m/s}$$ Average velocity = 0 m/s
  • d) Average speed (round trip):
    The total path length is 60 m + 60 m = 120 m. The total time is 10 s + 50 s = 60 s.
    $$\overline{v_s} = \dfrac{l}{\Delta t} = \dfrac{120}{60} = 2.0\text{ m/s}$$ Average speed = 2.0 m/s

Instantaneous Speed and Instantaneous Velocity

When we want to know how fast an object is moving at a specific, exact moment in time, we use instantaneous values.

Instantaneous Velocity (Vector): The average velocity measured over an extremely short, infinitesimally small time interval ($\Delta t \to 0$). On a position-time graph, this is exactly equal to the slope of the tangent line at that specific moment.

$$v = \lim_{\Delta t \to 0} \dfrac{\Delta x}{\Delta t} = \dfrac{dx}{dt}$$

Instantaneous Speed (Scalar): The magnitude (absolute value) of the instantaneous velocity. It is completely independent of the direction or the path taken.

$$v_s = |v|$$

Acceleration

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

Average Acceleration (Vector): The rate of change of velocity over a specific finite time interval. If an object accelerates from an initial velocity $v_0$ to a final velocity $v$ over a time interval $\Delta t$:

$$\overline{a} = \dfrac{\Delta v}{\Delta t} = \dfrac{v - v_0}{t - t_0}$$

Instantaneous Acceleration (Vector): The rate of change of velocity at a specific, exact moment in time. It measures the acceleration over an infinitesimally small time window ($\Delta t \to 0$). On a velocity-time graph, this corresponds exactly to the slope of the tangent line at that specific moment.

$$a = \lim_{\Delta t \to 0} \dfrac{\Delta v}{\Delta t} = \dfrac{dv}{dt}$$

Both parameters are measured in metres per second squared (m/s²).

⚠ Direction and Sign Conventions

Because velocity and acceleration 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 (m/s) 0 2 4 6 8
Acceleration (m/s²) - 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 (m/s) 0 2 6 10 18
Acceleration (m/s²) - 2 4 4 8