Strictly speaking…all motion is relative to something.

Usually that something is a reference point that is assumed to be at rest (i.e. the earth).

Motion can be relative to anything…even another moving object.

Relative motion problems involve solving problems with multiple moving objects which may or may not have motion relative to the same reference point. In fact, you may be given motion information relative to each other.

We use a combination of subscripts to indicate what the quantity represents and what it is relative to.

For example, “va/b” would indicate the velocity of “object a” with respect to “object b”. Object b in this example is the reference point. Note: The

“reference point” object is assumed to be at rest.

A plane flies due north with an airspeed of 50 m/s, while the wind is blowing 15 m/s due East. What is the speed and direction of the plane with respect to the earth?What do we know?

apv /

“Airspeed” means the speed of the plane with respect to the air.

“wind blowing” refers to speed of the air with respect to the earth.

eav /

What are we looking for? “speed” of the plane with

respect to the earth. epv /

We know that the speed and heading of the plane will be affected by both it’s airspeed and the wind velocity, so… just add the vectors.

eaapep vvv ///

eaapep vvv ///

So, we are adding these vectors…what does it look like? Draw a diagram,of the vectors tip to tail! Solve it!

smv ap /50/

N

smv ea /15/

epv /

θ

This one is fairly simple to solve once it is set up…but, that can be the tricky part. Let’s look at how the vector equation is put together and how it leads us to this drawing.

eaapep vvv ///

first last

middlesame

Note: We can use the subscripts to properly line up the equation. We can then rearrange that equation to solve for any of the vectors. Always draw the vector diagram, then you can solve for any of the vector quantities that might be missing using components or even the law of sines.

Other quantities can be solved for in this way, including displacement.

Remember that d=vt and so it is possible to see a problem that may give you some displacement information and other velocity information but not enough of either to answer the question directly

When solving these, be very careful that all the quantities on your diagram and in your vector formula are alike (i.e. all velocity or all displacement). Do not mix them!

If car A is moving 5m/s East and car B, is moving 2 m/s West, what is car A’s speed relative to car B.

So, we want to know…if we are sitting in car B, how fast does car A seem to be approaching us? Common sense tells us that Car A is coming at us at a rate of 7 m/s.

How do we reconcile that with the formulas?

2 m/s5 m/s

Car A Car B

Va/e =5 m/sCar A Car B

Vb/e = -2 m/s

Let’s start with defining the reference frame for the values given. Both cars have speeds given with respect to the earth.

If we set up the formula using the subscript alignment to tell us what to add, we get…

Eastsmv

vvv

ba

ebeaba

,/725/

///

We are looking for the velocity of A with respect to B, so va/b = ?

Then we need to solve for va/b .ebbaea vvv ///

So…