Speed, Velocity, Distance and Displacement

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The delta notation

With Δsometing  we mean a difference or change of that something.  If you had 15 marbles and you give away a few and end up having 10 of them, them

Δ(number of marbles)=15–10=5 marbles.

If the mass of a fruit is 120 g and then it dries down to 112 g, then

Δm = 120-112= 8 g

The notation Δm is simply read “delta em”.

Scalars vs Vectors

A  scalar is just a number and possibly a unit representing some physical quantity. Think of it as something that could be measured on a scale. Mass is an example of a physical quantity that is a scalar. A scalar will thus have a magnitude (size)  and possibly a unit, like 12 kg. A vector on the other hand has a direction too. A wind does not only blow with say 19 m/s, but also in a specific direction. In this case we often describe the direction by saying from what direction the wind is coming.  We may for example have a Northern wind at
19 m/s.

Most pre DP physics problems are  restricted to straight line problems, and in that case it is enough to show the direction along the line with a sign. Say “+” to the right and “–” to the left or we may choose to let “+” indicate the up direction and “–” the down direction.

We often show a vector using an arrow with the length proportional to the magnitude.  We can for example see this in some metrological charts.

Distance vs. Displacement

If an object move from one point in space to another, then the distance travelled is along its whole path. The distance will be a scalar, i.e. just a number, and in this case a non negative number.  The  displacement on the other hand is the straight line distance, and it will have a direction. It will thus be a vector.

Say for example we move an object from the point A to the point B along a somewhat crooked path  as in the figure. The length of the path along the way we actually move the object is say 14 cm, so the distance travelled is 14 cm. But say that the straight line distance from A to B is 1 cm, then the displacement is 1 cm to the right. If that is chosen to be our positive direction then the displacement is 1 cm. If we now move the object back again, then the new distance travelled will be 14 cm again and the displacement  will be –1 cm, i.e. 1 cm to the left.

The total distance travelled after these two movements will be 28 cm, and the displacement will be 0 cm, since we ended up where we started.

Speed vs. Velocity

Average speed and average velocity are both basically defined as

\bar v = \frac{{\Delta s}}{{\Delta t}}

Both speed and velocity is usually written with the letter v, but the velocity is sometimes written either in boldface italics,  v, or with an arrow above the letter, {\vec v} or on a blackboard or whiteboard with the so called black board bold where one do a vertical double stroke through the letter: All those way are to indicate that we have a vector, i.e. a direction.

  • The average speed is defined as the distance travelled divided by the time taken. It is a scalar, and always non negative, i.e. 0 or positive.
  • This is  also often written as change in distance divided by change in time.

If the movement from A to B  in the example above would take 7 seconds, then the average speed would be 14 cm / 7 s = 2 cm /s or 0.02 m/s.

A travel of  150 metres  in 15 seconds., would give an average speed of
150/15=10 m/s.

  • The average velocity  is defined as the displacement divided by the time taken.
  • This is often written as change in position divided by change in time.

The displacement is then the change in position.

If the movement from A to B  in the example above would take 7 seconds, then the average velocity  would be 1 cm / 7 s = 0.14 cm/s. In the opposite  direction it would be –0.14 cm/s.

To remember that velocity is the one with a direction you might think of the “v” in “velocity” as an arrow as in the arrow notation of vectors.

If we know the position at one time, say we are 10 m away from some particular place at the 4 s after midnight, and then, after seeing a frightening shadow at the particular place in question we find ourself  100 m away at 13 s after midnight, then our average speed was

\bar{v}=\frac{{{s}_{2}}-{{s}_{1}}}{{{t}_{2}}-{{t}_{1}}}=\frac{100-10}{13-4}=\frac{90}{9}=10\text{ m/s}

The speed (or velocity) is thus calculated as difference in position  (Δs) over difference in time (Δt) .

Units

In these pages we will (mostly) use the SI units. I.e. the the international system of units used by most countries in the world. In the US the United States customary units system is used.

The SI unit for distance and displacement is the metre (or meter) that is abbreviated as m. The unit for speed and velocity is m/s which is often written as ms-1.

We might sometimes, of convenience and/old habit  use other units, like cm/s or km/h.

We should have a space between the magnitude and the unit like in 12 m, and the unit should not be in italics. If you write 12m it might be mistaken for 12 times a mass (even though the abbreviation for mass, m, should be written using italics).

Up a level : Pre DP, Mechanics and Electronics
Next page : More on velocity and speedLast modified: Jan 27, 2017 @ 18:23