Kinematics Consolidation — Problem Solving & Exam Technique

Almost every Kinematics mistake begins in the first ten seconds, before any number is written — when a student fails to decide whether the question wants a scalar (size only) or a vector (size + direction), and fails to name a sign convention. Get these two decisions right and the maths usually follows; get them wrong and even perfect arithmetic earns nothing.

Use this two-question filter on every motion problem:

• Q1 — Does the answer need a direction? "How far is the car from the start?" needs a direction → displacement (vector). "How far did the car travel?" does not → distance (scalar). Same for speed (scalar) vs velocity (vector).
• Q2 — Which direction is positive? Write it down: "Take up = +" or "Take east = +". Every quantity then gets a sign and you can add with ordinary arithmetic instead of geometry.

Before calculating, make two decisions: (1) scalar or vector? — does the answer need a direction? (2) state a sign convention (e.g. up = +). For vertical motion, $a = -9.8$ m/s² throughout if up is positive. A declared convention turns a "lucky" sign into a defendable one.

Pause — copy the two-question filter and the "up = +, so $a=-9.8$ m/s²" sign rule into your book before moving on.

We just saw that direction and sign decisions come first. That raises a question: what happens when the data arrives not as words but as a graph the marker expects you to read in seconds? This card answers it → a fixed routine for extracting velocity, acceleration and displacement from any motion graph.

Graph questions feel slow only because students re-derive the rules each time. Lock in what each feature means and a graph becomes faster than a worded problem. There are just two graph types and two operations — gradient and area — to remember.

$x$–$t$ gradient = velocity. $v$–$t$ gradient = acceleration; area under a $v$–$t$ graph = displacement (areas below the axis are negative). Read the axis units before you calculate, and split awkward areas into triangles + rectangles.

Pause — copy the four graph reads (gradient $x$–$t$, gradient $v$–$t$, area $v$–$t$, sign of area) into your book before moving on.

We just saw how to pull velocity, acceleration and displacement straight off a graph. That raises a question: when there is no graph and the motion is uniformly accelerated, which of the five equations do you reach for? This card answers it → a single selection rule based on the variable you are not given.

For uniform acceleration there are five equations of motion (SUVAT). Each one is missing exactly one of the five variables $s, u, v, a, t$. The whole skill is: list what you know and what you want, then pick the equation that does not contain the variable you neither know nor want.

For uniform acceleration, list $s,u,v,a,t$ — note the one variable you neither know nor want, then pick the equation that omits it. $v=u+at$ (no $s$), $s=ut+\tfrac12 at^2$ (no $v$), $v^2=u^2+2as$ (no $t$), $s=\tfrac12(u+v)t$ (no $a$), $s=vt-\tfrac12 at^2$ (no $u$).

Pause — copy the five equations with the omitted-variable note beside each, and the four-step selection method, into your book before moving on.

We just saw the equation-selection rule for one-dimensional uniform acceleration. That raises a question: what changes when the motion leaves the straight line — a plane in a crosswind, a boat across a flowing river? This card answers it → a fixed vector-addition routine for relative motion on a plane.

Relative-motion problems trip students because they try to add speeds as numbers. The fix is to treat every velocity as a vector and combine them tip-to-tail. For a plane in wind or a boat in a current, the ground velocity is the vector sum of the craft's velocity (relative to the air/water) and the air/water's velocity (relative to the ground).

Relative/crosswind/river: ground velocity = vector sum of craft velocity + medium velocity. For perpendicular vectors, magnitude $=\sqrt{v_1^2+v_2^2}$ and direction $\theta=\tan^{-1}(v_2/v_1)$. "A relative to B" $=\vec{v}_A-\vec{v}_B$. Never add perpendicular speeds as plain numbers.

Pause — copy the resultant formula, the boat worked example, and "$\vec{v}_{AB}=\vec{v}_A-\vec{v}_B$" into your book before moving on.

We just saw the vector method for relative motion. That raises a question: if the methods are clear, why do marks still slip away in exams? This card answers it → because a small set of recurring errors strikes again and again. Name them and you can catch them in your own working.

Examiners report the same handful of kinematics mistakes every year. A "spot-the-error" skill is worth as much as a calculation skill — most band-6 students earn marks by checking, not just computing. Here is the error checklist to run over every answer before you move on.

Run the five-point check on every answer: (1) units consistent (km/h ÷ 3.6 = m/s)? (2) vector vs scalar — direction included where needed? (3) sign of $a$ matches your convention? (4) at the top, $v=0$ but $a=9.8$ down still? (5) $u=0$ for "from rest", $v=0$ for "stops"?

Pause — copy the five-point error checklist into your book; this is your final-check routine for every exam answer.