Kinematics of Uniform Rectilinear Motion

Kinematics is the branch of mechanics that describes how bodies move, deliberately setting aside why they move — that question belongs to dynamics, and to Newton's laws waiting a few chapters ahead. This chapter opens the topic with its simplest, most tractable case: uniform rectilinear motion, motion along a straight line at a velocity that never changes. "Rectilinear" means the path is a straight line; "uniform" means equal displacements occur in equal intervals of time, so the acceleration is identically zero throughout. Every idea introduced here — frames of reference, the distinction between distance and displacement, and between speed and velocity — is the scaffolding on which the rest of kinematics is built.

Deliberately absent from this chapter is any body whose velocity changes. Motion with non-zero, constant acceleration — the four SUVAT equations, free fall, and vertical throws — is the subject of the next chapter, and the full graphical machinery for reading $s$–$t$, $v$–$t$ and $a$–$t$ plots of any motion, uniform or not, follows in the chapter after that. Two-dimensional motion, including projectile motion, is developed several chapters further into this block, once the one-dimensional tools below are secure.

1. Position, Displacement and the Frame of Reference

Before any equation can be written, every kinematics problem requires a decision that is not, strictly speaking, physics at all: the choice of a frame of reference and a positive direction. Motion has no meaning in isolation — a passenger sitting still relative to a train is moving at $28~\dfrac{\text{m}}{\text{s}}$ relative to the track. Neither statement is "more true"; they are answers to different questions, and an IB solution is expected to state which frame is being used, usually implicitly through a sign convention (e.g. "east is positive").

With a frame of reference fixed, the position of a particle at time $t$ is described by a position vector, $\vec{r}(t)$. For motion confined to a straight line — the entirety of this chapter — position collapses to a single signed coordinate, $s(t)$, measured from a defined origin.

Displacement, $s$ (unit: $\text{m}$), is a vector quantity: the straight-line separation between an initial position and a final position, together with a direction (encoded, in one dimension, as a sign).

Displacement must not be confused with distance, $d$, a scalar that sums the entire length of the path actually travelled. An athlete completing one full lap of a $400~\text{m}$ circular track covers a distance of $400~\text{m}$, yet their displacement is exactly zero: the finish line is the start line. This distinction resurfaces constantly — average speed uses distance, average velocity uses displacement, and the two are only equal for motion in a single, unreversed direction, as Section 5 makes precise.

Key distinction. Distance and speed are scalars (magnitude only). Displacement, velocity and acceleration are vectors (magnitude and direction). In one-dimensional problems, the direction is carried entirely by the algebraic sign — a negative displacement is not an error, it is information.

2. Velocity and Speed

Velocity, $v$, is the rate at which displacement changes with time. Over a finite interval, the average velocity is the total displacement divided by the elapsed time:

In general — for a body whose velocity is changing — this single number discards almost everything of interest, since it says nothing about what happened between $t_1$ and $t_2$. The instantaneous velocity, the value read at one particular moment, is what a speedometer approximates, and for a general (non-uniform) motion the two can differ substantially. This chapter, however, is built around the one case where they cannot differ at all: uniform motion, defined and explored in Section 3.

Speed is simply the magnitude of velocity, $|v|$, stripped of its directional sign. A car's speedometer displays speed, never velocity — it has no way of knowing, or caring, which way the wheels are pointing. Speed is therefore always positive or zero, never negative, regardless of the sign carried by the corresponding velocity.

Key distinction. Velocity is a vector; speed is its magnitude, a scalar. A body moving at $v = -8.0~\dfrac{\text{m}}{\text{s}}$ has a speed of $8.0~\dfrac{\text{m}}{\text{s}}$ — the sign encodes direction only, and reporting a negative speed is never a valid IB answer.

3. Uniform Rectilinear Motion

A body undergoes uniform rectilinear motion when its velocity is constant: unchanging in both magnitude and direction. Because the velocity never changes, the instantaneous velocity at every instant equals the average velocity over any interval of the motion — there is no distinction left to draw between them, which is precisely what "uniform" means kinematically.

3.1 The equation of uniform motion, $s = s_0 + vt$

Let $s_0$ denote the position of the body at $t=0$. Because the velocity is constant, the displacement accumulated by any later time $t$ is simply that constant rate multiplied by the elapsed time, $\Delta s = v(t-0) = vt$. Since $\Delta s = s(t) - s_0$ by definition, rearranging gives the single equation that governs every problem in this chapter:

This is a linear relationship between $s$ and $t$: the position grows (or falls, if $v<0$) by equal amounts in equal time intervals, which is exactly the "equal distances in equal times" description of uniform motion. Equation (1) plays the same organising role in this chapter that the four SUVAT equations will play in the next — except here there is only one equation to remember, because acceleration never enters the picture.

Worked example. A cyclist passes a lamp post at $t=0$, already travelling at a constant $6.0~\dfrac{\text{m}}{\text{s}}$. Taking the lamp post as the origin, $s_0 = 0$ and $v = 6.0~\dfrac{\text{m}}{\text{s}}$, so

After $t = 5.0~\text{s}$ the cyclist is at $s(5.0) = 30~\text{m}$ past the post — found by nothing more than substitution, since the velocity never changes throughout the interval.

3.2 Motion in the negative direction and motion at rest

Equation (1) applies without modification when $v<0$: the body simply moves steadily in the negative direction, and $s(t)$ decreases linearly with time. The special case $v=0$ describes a body permanently at rest at $s=s_0$ — trivially "uniform," since a velocity that is constantly zero is still constant.

4. Interpreting $s$–$t$ Graphs of Uniform Motion

Equation (1), $s = s_0 + vt$, has exactly the form of a straight line, $y = c + mx$, with the gradient playing the role of velocity and the vertical intercept playing the role of the initial position:

• The gradient of the $s$–$t$ line equals the constant velocity, $v = \dfrac{\Delta s}{\Delta t}$, read between any two points on the line — the whole point of uniform motion is that this ratio does not depend on which two points are chosen.
• The vertical intercept, $s_0$, is the position at $t=0$.
• A steeper line corresponds to a larger speed; a line sloping downward (negative gradient) corresponds to motion in the negative direction; a horizontal line ($\text{gradient}=0$) corresponds to a body at rest.
• Because the gradient is constant, an $s$–$t$ graph of uniform motion is always a straight line — any curvature is immediate evidence that the velocity is changing, i.e. that the motion is not uniform, and that equation (1) no longer applies (see the next chapter for that case).

Worked example. The motion of a trolley is described by $s(t) = 12 + 4.0t$ (metres, seconds). Reading the equation directly against $s=s_0+vt$: the trolley starts $s_0 = 12~\text{m}$ from the origin and moves at a constant $v = 4.0~\dfrac{\text{m}}{\text{s}}$. In the first second its displacement changes by $s(1)-s(0) = 16-12 = 4.0~\text{m}$ — exactly $v$, as it must for every one-second interval of a uniformly moving body.

4.1 Two bodies on the same axes

When the $s$–$t$ graphs of two uniformly moving bodies are drawn on the same axes, their intersection marks the instant at which the two bodies occupy the same position — they meet (or pass) at that point. Two parallel lines, by contrast, share the same gradient and therefore the same velocity: the bodies move together, maintaining a constant separation, and never meet at all (unless they already coincide, in which case the two lines are identical). Distinguishing "same position" (an $s$–$t$ intersection) from "same velocity" (parallel $s$–$t$ lines, or an intersection on a velocity–time graph instead) is a frequent source of confusion, and is revisited with full graphical machinery two chapters on, in the chapter devoted to $s$–$t$, $v$–$t$ and $a$–$t$ graphs.

5. Average Speed and Average Velocity for Non-Uniform Paths

Not every journey is uniform for its entire duration — a car may cruise at one constant speed, then stop, then cruise again — yet the definitions of average speed and average velocity apply regardless, describing the journey's overall behaviour without needing to know the details of each individual stage.

Average speed is a scalar built from the total distance travelled — it can never be zero for a body that has actually moved, and it never distinguishes between forward and backward progress. Average velocity is a vector built from the net displacement — it depends only on where the journey started and ended, not on the path taken in between, and it can be zero even for a very long journey. The two coincide only when the motion never reverses direction, so that distance and the magnitude of displacement are equal throughout.

Worked example. A car travels $60~\text{km}$ due east in $1.0~\text{h}$, then returns along the same road to its starting point, taking a further $2.0~\text{h}$. Over the whole $3.0~\text{h}$ journey:

The average speed is $40~\dfrac{\text{km}}{\text{h}}$, found from the full $120~\text{km}$ actually driven; the average velocity is exactly zero, because the car ends the journey at the same position from which it started, so the net displacement vanishes. Notice that $40~\dfrac{\text{km}}{\text{h}}$ is not the simple arithmetic mean of the two leg speeds, $\tfrac{1}{2}(60+30)=45~\dfrac{\text{km}}{\text{h}}$ — that shortcut is only valid when both legs last for equal times, which they do not here, since the return leg (at $30~\dfrac{\text{km}}{\text{h}}$) takes twice as long as the outbound leg.

6. Relative Velocity in One Dimension

When two bodies both move, it is frequently useful to describe the motion of one relative to the other, rather than relative to the ground. For motion confined to a single line, the velocity of A relative to B is found by ordinary vector (here, signed) subtraction:

where $v_\text{A}$ and $v_\text{B}$ are both measured in the same fixed frame (typically the ground). Two immediate consequences follow directly from the sign of this subtraction:

• Same direction (chasing): if $v_\text{A}$ and $v_\text{B}$ have the same sign, the magnitude of the relative velocity is the difference of the two speeds — the gap between them closes (or opens) slowly.
• Opposite directions (approaching or separating): if $v_\text{A}$ and $v_\text{B}$ have opposite signs, the magnitude of the relative velocity is the sum of the two speeds — the gap changes quickly.

Worked example — passenger on a train. A train moves at a constant $30~\dfrac{\text{m}}{\text{s}}$ east relative to the ground. A passenger walks along the aisle at $1.5~\dfrac{\text{m}}{\text{s}}$ east relative to the train. Both velocities point the same way, so they add: the passenger's velocity relative to the ground is

Worked example — two trains passing. Two trains approach each other on parallel, straight tracks: train A moves east at $22~\dfrac{\text{m}}{\text{s}}$ and train B moves west at $18~\dfrac{\text{m}}{\text{s}}$. Taking east as positive, $v_\text{A} = +22~\dfrac{\text{m}}{\text{s}}$ and $v_\text{B} = -18~\dfrac{\text{m}}{\text{s}}$, so the velocity of A relative to B is

The trains close the gap between them at $40~\dfrac{\text{m}}{\text{s}}$ — the sum of their individual speeds, exactly as expected for two bodies approaching each other. If the trains are initially $2000~\text{m}$ apart, the time until they draw level is found from this relative velocity precisely as an ordinary uniform-motion problem, treating one train as stationary and the other as moving at the relative velocity:

Worked example — catching up. A cyclist $B$ is $80~\text{m}$ ahead of cyclist $A$ when both are moving in the same direction at constant speeds, $A$ at $8.0~\dfrac{\text{m}}{\text{s}}$ and $B$ at $5.0~\dfrac{\text{m}}{\text{s}}$. Since both velocities have the same sign, the relative velocity is the difference: $v_{\text{AB}} = 8.0-5.0 = 3.0~\dfrac{\text{m}}{\text{s}}$, the rate at which $A$ closes the $80~\text{m}$ gap. The time for $A$ to catch $B$ is $t = \dfrac{80}{3.0} \approx 27~\text{s}$.

7. Common Exam Pitfalls

• Confusing distance and displacement, and consequently average speed with average velocity. They coincide only when the motion never reverses direction.
• Averaging two speeds arithmetically when the two legs of a journey last for different times. Only $\bar{v}_{\text{speed}} = d/\Delta t$, built from the actual total distance and total time, is valid in general.
• Reading the gradient of an $s$–$t$ graph as if it were a position, or the intercept as if it were a velocity. The gradient is always $\Delta s/\Delta t$ (a velocity); the intercept is always the value of $s$ at $t=0$.
• Treating a curved $s$–$t$ graph as if it still represented uniform motion. Only a straight line has a constant gradient; a curve signals a changing velocity, which places the problem outside the scope of equation (1) and this chapter.
• Getting the sign wrong in relative velocity. Subtracting when the two velocities point in opposite directions (or adding when they point the same way) inverts the physical result; always fix a sign convention first and check whether the two velocities being combined share a sign or not.
• Mixing units. Equation (1) requires SI base units throughout ($\text{m}$, $\dfrac{\text{m}}{\text{s}}$, $\text{s}$); a velocity quoted in $\dfrac{\text{km}}{\text{h}}$ must be converted before substitution.

Summary

• Displacement $s$ is a vector (straight-line change of position); distance $d$ is a scalar (total path length travelled), and $d \ge |\Delta s|$ always, with equality only for unreversed straight-line motion.
• Velocity is a vector; speed $|v|$ is its magnitude, always non-negative.
• Uniform rectilinear motion is motion at constant velocity: equal displacements in equal times, zero acceleration, and instantaneous velocity always equal to average velocity over any interval.
• The single equation governing this chapter is $s = s_0 + vt$, a straight line on an $s$–$t$ graph whose gradient is the velocity and whose intercept is the initial position.
• Average speed $= d/\Delta t$ (scalar, uses total distance) and average velocity $= \Delta s/\Delta t$ (vector, uses net displacement) generally differ for a journey that reverses direction, even though both remain well defined.
• Relative velocity in one dimension is found by signed subtraction, $v_{\text{AB}} = v_\text{A} - v_\text{B}$: same-direction motion subtracts speeds, opposite-direction motion adds them.
• Motion with a changing velocity — the four SUVAT equations, free fall, and full graphical analysis of $s$–$t$, $v$–$t$ and $a$–$t$ plots — is developed in the next two chapters; two-dimensional motion, including projectile motion, follows several chapters after that.