Relative Motion in Two Dimensions

Our two observers are again watching a moving particle P from the origins of reference frames A and B, while B moves at a constant velocity

relative to A. (The corresponding axes of these two frames remain parallel.) Figure 4-22 shows a certain instant during the motion. At that instant, the position vector of the origin of B relative to the origin of A is

. Also, the position vectors of particle P are

relative to the origin of A and

relative to the origin of B. From the arrangement of heads and tails of those three position vectors, we can relate the vectors with

(4-43)

By taking the time derivative of this equation, we can relate the velocities

and

of particle P relative to our observers:

(4-44)

By taking the time derivative of this relation, we can relate the accelerations

and

of the particle P relative to our observers. However, note that because

is constant, its time derivative is zero. Thus, we get

(4-45)

As for one-dimensional motion, we have the following rule: Observers on different frames of reference that move at constant velocity relative to each other will measure the same acceleration for a moving particle.

FIGURE 4-22

Frame B has the constant two-dimensional velocity

relative to frame A. The position vector of B relative to A is

. The position vectors of particle P are

relative to A and

relative to B.

In Fig. 4-23a, a plane moves due east while the pilot points the plane somewhat south of east, toward a steady wind that blows to the northeast. The plane has velocity

relative to the wind, with an airspeed (speed relative to the wind) of 215 km/h, directed at angle

south of east. The wind has velocity

relative to the ground with speed of 65.0 km/h, directed 20.0° east of north. What is the magnitude of the velocity

of the plane relative to the ground, and what is

FIGURE 4-23

A plane flying in a wind.

Calculations: First we construct a sentence that relates the three vectors shown in Fig. 4-23b:

This relation is written in vector notation as


	(4-46)

We need to resolve the vectors into components on the coordinate system of Fig. 4-23b and then solve Eq. 4-46 axis by axis. For the y components, we find

Solving for

gives us


	(Answer)

Similarly, for the x components we find

Here, because

is parallel to the x axis, the component

is equal to the magnitude

. Substituting this notation and the value

, we find


	(Answer)