Kinematics of linear motion, explained simply

Kinematics of linear motion is a favourite of many students, because it takes the calculus you already learned and attaches it to a picture: a single particle moving back and forth along a straight line, measured from a fixed point $O$. Everything in the chapter is really about three quantities and the links between them.

• Displacement $s$: how far, and in which direction, the particle is from $O$. It can be positive or negative.
• Velocity $v$: how fast the displacement is changing, with direction. Its sign tells you which way the particle is moving.
• Acceleration $a$: how fast the velocity is changing.

The link is calculus. To go down the chain, from displacement to velocity to acceleration, you differentiate with respect to time $t$:

$$v = \frac{ds}{dt}, \qquad a = \frac{dv}{dt} = \frac{d^{2}s}{dt^{2}}$$

To go back up the chain, from acceleration to velocity to displacement, you integrate, and each integration brings an unknown constant that a given condition (like "the particle starts at $O$") lets you find:

$$v = \int a\, dt, \qquad s = \int v\, dt$$

The single biggest source of lost marks in kinematics is not the calculus, it is the signs and the words. A negative velocity does not mean "slowing down"; it means moving in the negative direction.

Speed is the size of the velocity, $|v|$, and is never negative. Getting these words right is half the chapter.

Three moments matter, and each is spotted by setting something to zero:

• The particle is momentarily at rest when $v = 0$. This is where it can change direction.
• The particle is back at the starting point $O$ when $s = 0$ (if $s$ is measured from $O$).
• The velocity is at a maximum or minimum when $a = \dfrac{dv}{dt} = 0$.

This distinction is tested constantly, so it is worth pinning down. Displacement over an interval is just the final position minus the initial position, it can be negative, and it ignores any wandering in between.

Distance travelled is the total length of the path, so it counts every metre, even when the particle doubles back.

The reason they differ is direction changes. If the particle never changes direction in your interval, distance and the size of the displacement are the same.

If it does change direction, that is, if $v = 0$ somewhere inside the interval, you must split the interval at that moment and add the distances of each leg separately.

Suppose a particle moves along a straight line so that its displacement from $O$, in metres, is given by the function below, where $t$ is the time in seconds.

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

Differentiate once for velocity, and again for acceleration:

$$v = \frac{ds}{dt} = 3t^{2} - 12t + 9 = 3(t-1)(t-3)$$

$$a = \frac{dv}{dt} = 6t - 12 = 6(t-2)$$

The velocity is zero when $t = 1$ and $t = 3$, so the particle is momentarily at rest at those times, the moments it changes direction. The acceleration is zero at $t = 2$, which is where the velocity reaches its least value.

Now find the total distance travelled in the first 3 seconds. Because the particle turns at $t = 1$, split the journey there.

The positions are $s(0) = 0$, $s(1) = 4$ and $s(3) = 0$:

• From $t = 0$ to $t = 1$: position changes from $0$ to $4$, a distance of $4$ m.
• From $t = 1$ to $t = 3$: position changes from $4$ back to $0$, a distance of $4$ m.

Total distance $= 4 + 4 = 8$ m, even though the displacement over the whole interval is $0$, the particle ended exactly where it began. That gap between $8$ m and $0$ m is the whole point of the distinction.

Kinematics rewards a student who keeps the chain, differentiate down, integrate up, and the sign conventions crystal clear, and it quietly punishes anyone who blurs them. Working one-to-one, a teacher can watch exactly where your reasoning slips: whether it is a sign, a misread word like "at rest", or forgetting to split the interval for distance.

Fixing the specific gap is far faster than re-reading the whole chapter. Our teachers are experienced; lessons are online and taught in English, while the SPM papers are set bilingually in Bahasa Melayu and English.

If kinematics is where your marks leak, you are welcome to begin with a one-hour paid trial class at the teacher's own rate (from RM50 per hour, depending on experience). We won't promise a grade, but we can help you make the chain, and the signs, feel automatic.