Speed and Velocity, Rates of Motion

"How fast?" and "how fast, and which way?" are two different questions. A pedestrian sign saying "40 m/s" tells you a rate distance covered per second, but says nothing about direction. The moment you add "east", you have answered a richer question. That extra word is the entire difference between speed and velocity.

Speed is a scalar: it measures how fast distance is being covered, with no direction. Velocity is a vector: it measures the rate of change of position (displacement), so it carries a direction. Speed pairs with distance (both scalars); velocity pairs with displacement (both vectors).

Same number, different meaning

Speed is a scalar (how fast, no direction) and pairs with distance. Velocity is a vector (how fast AND which way) and pairs with displacement. In 1D, velocity carries a + or − sign; +30 m/s and −30 m/s are the same speed but opposite velocities.

Pause, copy the speed vs velocity definitions and the rule "speed↔distance, velocity↔displacement" into your book before moving on.

We just saw that speed pairs with distance and velocity pairs with displacement. That raises a question: how do we put numbers on these rates over a whole journey? This card answers it → with the average formulas, where the only difference is whether you divide distance or displacement by time.

An average compresses a whole journey into one rate. To get it, you take the total amount of motion and share it evenly across the total time, as if the object had moved steadily the entire way, even though it usually did not.

Average speed uses the total distance (a scalar). Average velocity uses the total displacement (a vector), so it keeps a direction:

Here $\Delta x$ ("delta x") means the change in position, final position minus initial position, and $\Delta t$ is the time interval. Because displacement can be smaller than distance (or even zero), average velocity can be smaller than average speed for the same trip.

Average speed $= d/t$ (total distance over time). Average velocity $\vec{v}_{\text{av}} = \Delta x/\Delta t$ (displacement over time, with direction). For a one-way straight trip they have equal magnitude; if the object backtracks, displacement < distance so average velocity < average speed.

Pause, copy both average formulas and the out-and-back jogger example (speed 8 km/h, velocity 0 km/h) into your book before moving on.

We just saw how to find the average velocity over a whole trip. That raises a question: the speedo doesn't show the trip average, it shows what's happening right now, so how do we measure the velocity at a single instant? This card answers it → by shrinking the time interval until it is almost zero.

Your car's speedometer does not wait until the end of the journey to report. It tells you how fast you are going at this moment. That moment-by-moment value is the instantaneous velocity and it is what changes second to second as you accelerate, brake or turn.

Instantaneous velocity is the velocity at one particular instant. We find it by measuring displacement over a very small time interval, the smaller $\Delta t$, the closer the result is to the true value at that instant:

The average velocity, by contrast, uses the whole trip. If the velocity never changes, the instantaneous and average values are identical. But whenever the motion speeds up or slows down, the instantaneous value drifts above or below the average.

Instantaneous velocity is the velocity at one instant, found from $\Delta x/\Delta t$ over a very small $\Delta t$. Average velocity uses the whole interval. They are equal only when velocity is constant; otherwise the instantaneous value rises above or falls below the average.

Pause, copy the definition of instantaneous velocity and the "smaller $\Delta t$ = closer to the true instant" idea into your book before moving on.

We just saw how to find velocity over both long and tiny time intervals. That raises a question: the speedo reads km/h but our formulas give m/s, so how do we move between the two units without error? This card answers it → with the factor of 3.6 that links m/s and km/h.

Physics calculations use the SI unit metres per second (m/s), but everyday speeds, and your speedometer, use kilometres per hour (km/h). You must be able to swap between them, because exam data often mixes the two.

There are 1000 m in a kilometre and 3600 s in an hour. Combining these gives a single conversion factor of 3.6:

So 10 m/s $\times 3.6 = 36$ km/h, and 108 km/h $\div 3.6 = 30$ m/s. A handy benchmark: a relaxed walk is about 1.5 m/s ($\approx$ 5 km/h), and the urban speed limit of 50 km/h is about 14 m/s.

To convert speed: m/s $\times 3.6 =$ km/h, and km/h $\div 3.6 =$ m/s. The 3.6 comes from 3600 s/h ÷ 1000 m/km. Always convert everything to consistent units (usually m/s) before substituting into a formula.

Pause, copy the conversion rule (×3.6 and ÷3.6) and the worked conversions (108 km/h = 30 m/s) into your book before moving on.

We just saw how to handle the units of velocity. That raises a question: in a real lab, how do we actually gather data to find both the average and the instantaneous velocity of a moving object? This card answers it → with three standard technologies: the ticker timer, the data logger, and video analysis.

The syllabus asks you to conduct a practical investigation to gather data to analyse instantaneous and average velocity. The trick is always the same: record an object's position at known, regular time intervals, then use $\vec{v} = \Delta x/\Delta t$, over the whole run for the average, and over one tiny interval for the instantaneous value.

Three technologies do this job:

• Ticker timer. A vibrating arm punches dots on a paper tape pulled by the moving trolley, typically 50 dots per second (so each gap = $\tfrac{1}{50}=0.02$ s). Close dots mean slow; widely spaced dots mean fast. Measuring one gap gives an instantaneous velocity; measuring the whole tape gives the average.
• Data logger with a motion sensor. An ultrasonic or light-gate sensor records position or time automatically, hundreds of times a second, straight to a computer, far more precise than reading a tape by hand.
• Video analysis. Film the motion against a measuring scale, then step through the frames (e.g. 30 frames per second, so $\Delta t = \tfrac{1}{30}$ s per frame). The position in each frame gives a position–time table.

To gather velocity data, record position at regular known time intervals using a ticker timer (50 dots/s, $\Delta t = 0.02$ s), a data logger (motion/light-gate sensor), or video analysis (frames at known fps). Use one tiny gap for instantaneous velocity and the whole run for average velocity, both from $\vec{v}=\Delta x/\Delta t$.

Pause, copy the three technologies and the ticker-tape method ($\Delta t = 1/50 = 0.02$ s; one gap = instantaneous, whole tape = average) into your book before moving on.