Motion in One Dimension

Constant Velocity

distance = speed x time

A cart moving at a steady speed travels a distance proportional to time. We track the units at every step and watch equal seconds map to equally spaced positions.

Example

A cart moving at a steady speed covers distance in proportion to time. We track the units and watch equal seconds map to equal spacing. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Read the motion

A cart moves at a constant speed of 3 metres per second for 4 seconds. We want the distance it travels.

v=3 m/s,t=4 sv = 3\ \text{m}/\text{s}, \quad t = 4\ \text{s}
A cart at the start of its motionA cart sits on a level track at the starting line with a velocity arrow pointing in the direction of motion.cartv

Choose the relationship

At constant velocity, distance is velocity multiplied by time. Check the units first: metres per second times seconds leaves metres, which is a distance.

x=v⋅tx = v \cdot t

Substitute the numbers

Put in the speed 3 and the time 4.

x=3 m/s ⋅ 4 sx = 3\ \text{m}/\text{s} \,\cdot\, 4\ \text{s}

Compute the distance

Multiplying gives 12 metres. Because the speed never changes, equal time steps cover equal distances.

x=12 mx = \hl{12}\ \text{m}
Equal time steps, equal spacing, equal arrowsSnapshots at each whole second sit at equally spaced marks with equal velocity arrows, because the speed never changes.t=0t=1t=2t=3cartv

Fixed time makes distance track speed

Hold the time at 4 seconds. Each bigger speed covers more distance in that same time.

vtx1 m/s4 s4 m2 m/s4 s8 m3 m/s4 s12 m\begin{array}{c|c|c}v & t & x \\ \hline 1\ \text{m}/\text{s} & 4\ \text{s} & 4\ \text{m} \\ 2\ \text{m}/\text{s} & 4\ \text{s} & 8\ \text{m} \\ 3\ \text{m}/\text{s} & 4\ \text{s} & 12\ \text{m}\end{array}

Fixed speed makes distance track time

Hold the speed at 3 metres per second. More seconds add more equal chunks of distance.

vtx3 m/s2 s6 m3 m/s3 s9 m3 m/s4 s12 m\begin{array}{c|c|c}v & t & x \\ \hline 3\ \text{m}/\text{s} & 2\ \text{s} & 6\ \text{m} \\ 3\ \text{m}/\text{s} & 3\ \text{s} & 9\ \text{m} \\ 3\ \text{m}/\text{s} & 4\ \text{s} & 12\ \text{m}\end{array}
mechanics The numbers here are deliberately small and clean so every step — including the unit check that turns metres-per-second times seconds into metres — can be read off exactly.