11A: Relative Velocity

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Vectors add like vectors, not like numbers. Except in that very special case in which the vectors you are adding lie along one and the same line, you can’t just add the magnitudes of the vectors.

Imagine that you have a dart gun with a muzzle velocity of \(45 mph\). Further imagine that you are on a bus traveling along a straight highway at \(55 mph\) and that you point the gun so that the barrel is level and pointing directly forward, toward the front of the bus. Assuming no recoil, as it leaves the muzzle of the gun, how fast is the dart traveling relative to the road? That’s right! \(100 mph\). The dart is already traveling forward at \(55 mph\) relative to the road just because it is on a bus that is moving at \(55 mph\) relative to the road. Add to that the velocity of \(45 mph\) that it acquires as a result of the firing of the gun and you get the total velocity of the dart relative to the road. This problem is an example of a class of vector addition problems that come under the heading of “Relative Velocity.” It is a particularly easy vector addition problem because both velocity vectors are in the same direction. The only challenge is the vector addition diagram, since the resultant is right on top of the other two. We displace it to one side a little bit in the diagram below so that you can see all the vectors. Defining

\(\vec{V}_{BR}\) to be the velocity of the bus relative to the road,
\(\vec{V}_{DB}\) to be the velocity of the dart relative to the bus, and

\(\vec{V}_{DR}\) to be the velocity of the dart relative to the road; we have

The vector addition problem this illustrates is
\[\vec{V}_{DR}=\vec{V}_{BR}+\vec{V}_{DB} \nonumber \]

If we define the forward direction to be the positive direction,

then, because the vectors we are adding are both in the same direction, we are indeed dealing with that very special case in which the magnitude of the resultant is just the sum of the magnitudes of the vectors we are adding:

\[\vec{V}_{DR}=\vec{V}_{BR}+\vec{V}_{DB} \nonumber \]

\[V_{DR}=V_{BR}+V_{DB} \nonumber \]

\[V_{DR}=55mph+45mph \nonumber \]

\[V_{DR}=100mph \nonumber \]

\(\vec{V}_{DR}=100mph\) in the direction in which the bus is traveling

You already know all the concepts you need to know to solve relative velocity problems (you know what velocity is and you know how to do vector addition) so the best we can do here is to provide you with some more worked examples. We’ve just addressed the easiest kind of relative velocity problem, the kind in which all the velocities are in one and the same direction. The second easiest kind is the kind in which the two velocities to be added are in opposite directions.