How Does the Vector Triangle Principle Apply to Different Physical Quantities?

[Greg Bernhardt]

Definition/Summary

Vectors (such as velocity or force or momentum) obey the Vector Law of Addition.

That means in particular that combination of two vectors $\vec{V}_1$ and $\vec{V}_2$ is a vector $\vec{V}_3$ only if three directed lines (lines with arrows) $\vec{L}_{AB}$ $\vec{L}_{BC}$ and $\vec{L}_{AC}$ can be drawn representing the vectors so as to make a closed triangle with the arrows going from A to B, from B to C, and from A to C.

Similarly, three vectors will add to zero only if three directed lines representing them can be drawn so as to make a closed triangle with the arrows going the same way round the triangle.

(And similarly for n vectors, and closed n-sided polygons.)

Pseudovectors (such as angular momentum) also obey the Vector Law of Addition, in exactly the same way as vectors.

Equations

$$\vec{V}_{AB}\ +\ \vec{V}_{BC}\ =\ \vec{V}_{AC}$$

$$\vec{V}_{AB}\ +\ \vec{V}_{BC}\ +\ \vec{V}_{CA}\ = 0$$

Extended explanation

Velocities:

Remember: technically, there is no such thing as an absolute velocity.

All velocities are relative velocities, between one object (or observer) and another.

Very often, one of those objects is the ground!

To make sure that the arrows are the right way round, give each relative velocity two letters, representing the two objects whose relative velocity it represents.

For example, $\vec{V}_{WB}$ and $\vec{V}_{GW}$ for "Water to Boat" and "Ground to Water", representing the velocity of a boat relative to the water, and of the water relative to the ground.

Then:

$$\vec{V}_{GW}\ +\ \vec{V}_{WB}\ =\ \vec{V}_{GB}$$

Similarly, you might use Ground Wind and Plane:

$$\vec{V}_{GW}\ +\ \vec{V}_{WP}\ =\ \vec{V}_{GP}$$

Forces:

When the change of momentum (acceleration) of a body is zero, Newton's second law means that the vector sum of the forces on that body is zero.

So a vector triangle (or polygon) may be used.

(The difference between a vector triangle (or polygon) and an FBD, or "Free-Body Diagram", is that the vectors in an FBD are all drawn starting at the same point, and Cartesian coordinates are used to determine that they add to zero.)

Before you draw any vector triangle for forces:

i] decide which body the forces are acting on (it must always be the same body)

ii] then draw extra lines with arrows on the diagram of the actual situation, to show the forces on that body.

iii] then draw the vector triangle (or polygon) separately, taking care that the arrows go round the triangle (or polygon) the same way!

Momenta:

In a collision, we treat all the bodies involved as being one (non-rigid) body.

Then there are no (external) forces on that one body, and so Newton's second law means that the change of momentum of that body is zero.

And therefore the vector sum of the individual momenta making up that body is zero.

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[Radhakrishnam]

Do binary oblique elastic collisions lead to velocity triangles in all cases, just as they lead to momenta triangles? Or, are there exceptions where the velocities do not form a triangle while the momenta do?