Kinematics: the maths of motion

Kinematics in Add Math is really an application chapter, nearly everything in it reuses differentiation and integration from earlier chapters, applied to a particle moving along a straight line. The whole topic rests on one relationship: displacement, velocity, and acceleration are three functions of time, and each one is found from the one before it by differentiating, or the one after it by integrating.

$$v = \frac{ds}{dt}, \qquad a = \frac{dv}{dt}$$

Read left to right, differentiating displacement $s$ with respect to time gives velocity $v$; differentiating velocity gives acceleration $a$. Read the other way, integrating acceleration gives velocity (plus a constant, found from given conditions), and integrating velocity gives displacement (plus another constant).

Once this chain is comfortable, most kinematics questions become a matter of deciding which direction along the chain the question requires, rather than learning a new technique for each one.

• Displacement $s$: position relative to a fixed starting point, measured with direction, it can be negative.
• Velocity $v$: rate of change of displacement, also signed, since direction still matters.
• Acceleration $a$: rate of change of velocity, positive acceleration does not always mean speeding up, if velocity is negative.

The single idea that separates confident kinematics answers from shaky ones is treating displacement, velocity, and acceleration as signed quantities along a line, not as plain sizes. A particle can have negative velocity while moving in the direction defined as negative, and a negative acceleration can mean speeding up if the velocity is also negative, the object is moving faster in the negative direction, not slowing down.

A common early confusion is assuming "the particle stops" and "the particle changes direction" are the same statement. They usually coincide, but not always, checking the sign of velocity just before and just after $v=0$ is the reliable way to tell whether the particle actually reverses or merely slows to zero and continues the same way, which happens, for example, at a point of inflexion in velocity.

This distinction accounts for a large share of the marks lost in kinematics, and it is worth treating as its own skill rather than an afterthought. Displacement over an interval is simply the difference between the final and initial position, direction and any doubling-back cancel out.

Total distance travelled adds up every bit of movement, regardless of direction, so any time the particle reverses direction within the interval, distance and displacement give different numbers.

The reliable method is to find every value of $t$ within the interval where $v=0$, since those are the only points where a reversal can happen. Splitting the interval at each of those points, working out the displacement over each smaller piece separately, and adding the absolute value of each piece gives total distance, while simply adding the pieces without the absolute value, letting negative displacements cancel positive ones, gives net displacement.

A particle moves in a straight line so that its velocity, in $\text{m s}^{-1}$, at time $t$ seconds is given by $v=3t^{2}-12t+9$, for $t\geq0$, starting from $s=0$ when $t=0$.

1. Find when the particle is momentarily at rest, by setting $v=0$: $3t^{2}-12t+9=0$, which factorises to $3(t-1)(t-3)=0$, giving $t=1$ and $t=3$.
2. Find acceleration by differentiating velocity: $a=\frac{dv}{dt}=6t-12$.
3. Find displacement by integrating velocity: $s=t^{3}-6t^{2}+9t+c$; since $s=0$ when $t=0$, $c=0$, giving $s=t^{3}-6t^{2}+9t$.

$$s = t^{3} - 6t^{2} + 9t$$

At $t=1$, $s=1-6+9=4$; at $t=3$, $s=27-54+27=0$. Between $t=0$ and $t=1$ the particle moves 4 m in the positive direction; between $t=1$ and $t=3$ it reverses and returns 4 m back to $s=0$.

The net displacement over the first 3 seconds is $0$ m, but the total distance travelled is $4+4=8$ m, a direct illustration of why the two are not interchangeable.

Kinematics leans heavily on calculus skills learned earlier, so slips here are often carried over from those chapters rather than new to this one.

• Forgetting the constant of integration when moving from acceleration to velocity, or velocity to displacement, kinematics questions almost always give a starting condition specifically to fix that constant.
• Reading a negative velocity as "the particle has stopped" rather than "the particle is moving in the negative direction."
• Calculating displacement over an interval and reporting it as total distance, without checking for a direction reversal first.
• Assuming maximum velocity happens where acceleration is zero without confirming it is a maximum rather than a minimum.

Kinematics sits at the intersection of several earlier chapters, differentiation, integration, and quadratic equations all show up inside a single question, so a shaky patch in any one of them can surface here even when the kinematics idea itself is understood. A teacher working with you one-to-one can trace a wrong answer back to which underlying skill actually needs attention, rather than re-teaching the whole chapter from scratch.

Our teachers are experienced; lessons run online and are taught in English, while SPM papers themselves are set bilingually in Bahasa Melayu and English.

If kinematics, or the calculus chapters it draws on, is a sticking point, a one-hour paid trial class at the teacher's own rate (from RM50 per hour, depending on experience) is a reasonable place to start.