Relative velocity and closing speed

Relative velocity and its components

Suppose we have objects A and B [1] with velocity vectors and . The relative velocity of B w.r.t. A is:

Put differently, it’s the velocity of B in A’s frame of reference.

This relative velocity is a vector, and we can split it into orthogonal components. Obviously, the nature of such a split depends on the basis we want to use. We could look at the vector’s x an y components (we’ll be using - two dimensional space, but everything here applies to 3D as well), but for this post we’re interested in something slightly different:

We draw a line connecting the two objects. The component of in the direction of this line is called the normal component of relative velocity, while the component perpendicular to this direction is called the tangential component.

Computing the normal component

How do we find the component of a vector in the direction of a specific line? By using a vector projection!. We’ll represent the line by a vector, and find the projection of onto this vector.

The positions of A and B can also be seen as vectors: and . The line connecting them can then be expressed as the vector :

All we need from this position difference vector is its direction, not its magnitude, however [2]. So we’ll use the unit vector of , denoted as:

Finally, to find the projection of onto , we compute [3]:

Where the multiplication operator between the vectors is the dot product. Note that the result of the dot product is a scalar; therefore, the quantity is called the closing speed - it expresses the rate at which the relative distance of the two objects is changing. If it’s positive, the objects are drifting farther apart; if it’s negative, the objects are getting closer together. Therefore the term "closing speed" may be slightly confusing; alternatively, this has been called a "signed separation speed", or "normal relative speed" [4].

The signs in these calculations can be tricky to get right, so we have to be very careful. Let’s see a few examples that will help us make these computations more concrete.

Examples

To build up some intuition and get some practice with the equations, we’ll review the following examples:

Example I: We’ll start by computing the relative position unit vector :

Then, the closing speed is:

Based on our convention, the negative sign of means that the objects are approaching each other. Due to the simple nature of the example, this result is easy to verify, as it can be immediately guessed just by looking at the diagram.

Example II: Here is the same as in the previous example. The closing speed is:

Same magnitude, but different sign from before, because the objects are moving farther apart.

Example III This example is to demonstrate that we get consistent results even if B is to the left of A. Here the relative position unit vector is:

And the closing speed:

Which is the same as in example I, as expected. The direction of flipped, but so did the direction of the relative velocity vector, so the result has the same sign.

Example IV: Finally, an example showing more arbitrary positions and velocities.

Then:

This example is a good opportunity to demonstrate something important about : it’s time-dependent, because positions change with time. Here, -6.6 is the closing speed at the exact moment when A’s and B’s positions and velocities are as stated in the example. In the next time step, the position of A will be and the position of B will be , while their velocities remain the same. The then will be quite different. This is a good segue to the next topic - which is a more physical view of closing speed.

Closing speed as a function of time

The computation shown so far represents a static view of the world; perhaps the right word to use is instantaneous. Given the positions and velocities at a given moment, what is the closing speed between the objects at that exact moment?

But there’s no reason to not generalize this using a more standard physical interpretation of velocity.

First, let’s state the position vectors of A and B as a function of time: and . The relative position vector between the objects is also a function of time:

Now we’ll define the scalar distance as the magnitude of this vector:

We’re interested in - the change in this distance over time. Let’s start by breaking down to its constituents:

By the chain rule [5]:

Switching back to the vector representation: since , the numerator of the fraction above is then a dot product between and , we can write this as:

But is precisely from the earlier section, as a function of time. Moreover:

Because velocity is the time derivative of position. Therefore, we end up with the same equation, just as a function of time:

This formulation is more precise because it makes it very obvious that all the quantities we’re dealing with are time dependent.