Kinematics in One Dimension: Worked Reasoning

≈ 25 min

Worked Reasoning

Make each reasoning step visible: identify the knowns, choose the governing idea, apply it and check the result.

This extension applies that lens specifically to Kinematics in One Dimension.

Kinematics describes motion without asking what causes it. Along a single straight line we track position $x(t)$ , and define

v = \frac{dx}{dt}, \qquad a = \frac{dv}{dt} = \frac{d^2x}{dt^2}

Velocity is the rate of change of position; acceleration is the rate of change of velocity. Both are signed: the sign records direction along the chosen axis, so you must choose a positive direction before you substitute a single number.

When $a$ is constant, integrating twice gives the three equations of motion:

v = v_0 + at \qquad \Delta x = v_0 t + \tfrac{1}{2}at^2 \qquad v^2 = v_0^2 + 2a\,\Delta x

The third is the first two with $t$ eliminated — useful whenever time is neither given nor wanted.

Worked example. A minibus taxi pulls away from a rank from rest with a constant acceleration of $2.5\ \text{m/s}^2$ for $8.0\ \text{s}$ .

Speed after $8.0$ s:

v = v_0 + at = 0 + (2.5)(8.0) = 20\ \text{m/s}

Distance covered:

\Delta x = v_0 t + \tfrac{1}{2}at^2 = 0 + \tfrac{1}{2}(2.5)(8.0)^2 = 80\ \text{m}

A cross-check with the third equation: $v^2 = 0 + 2(2.5)(80) = 400$ , so $v = 20\ \text{m/s}$ — consistent.

Physics — Year 1 — Worked Reasoning: A bakkie accelerates uniformly from rest to $20\ \text{m/s}$ in $8.0\ \text{s}$ . What is its acceleration, in $\text{m/s}^2$ ?

Physics — Year 1 — Worked Reasoning: On a velocity–time graph, the area between the graph and the time axis represents which quantity?

The displacement of the object

The acceleration of the object

The average speed of the object

The rate of change of acceleration

Physics — Year 1 — Worked Reasoning: A stone is dropped from rest off a bridge over the Vaal River and reaches the water $2.0\ \text{s}$ later. Taking $g = 9.8\ \text{m/s}^2$ and ignoring air resistance, how far did it fall, in metres?

Name the original topic being extended by this worked reasoning lesson.

Which statement is the best evidence-led starting point for Kinematics in One Dimension: Worked Reasoning?

Kinematics describes motion without asking what causes it. Along a single straight line we track position $x(t)$ , and define

v = \frac{dx}{dt}, \qquad a = \frac{dv}{dt} = \frac{d^2x}{dt^2}

Velocity is the rate of change of position; acceleration is the rate of change of velocity. Both are signed: the sign records direction along the chosen axis, so y

Newton's Laws and Free-Body Diagrams

Work, Energy and Power

A conclusion needs no observations, calculations or stated assumptions.