As you push the demonstration table across the room, you can give the small cart on the track on top of the table a motion relative to the table, and thus a different motion, relative to the class, from that of the table.

With this demonstration you can illustrate the concept of a frame of reference, and a variety of motions for the cart on the track, relative to the table and to the lecture hall. All the other demonstrations in this chapter illustrate the motion of some object, for example an air track glider or a ball, relative to whatever apparatus is used in the demonstration. Some of these demonstrations illustrate one-dimensional motion, and some illustrate two-dimensional motion, but for all of them, the apparatus itself is stationary. (The one possible exception is demonstration 08.24 -- Ball over tunnel, in which the track is stationary, but the cart undergoes one-dimensional motion while the ball that is launched from it undergoes two-dimensional motion.) The motion for the object is thus the same relative to both the apparatus and the lecture hall. For this demonstration, however, you move the rolling table, so that the motion of the cart on the track that is resting on the table, can be different relative to the table, from its motion relative to the lecture hall.

If we follow the convention used in demonstration 04.12 -- 3-D vector board, we can imagine that the lecture hall is aligned with a pair of coordinate axes, with the x-axis running across the width of the hall, the positive x direction toward audience right, and the y-axis running the length of the hall, the positive y direction toward the back. The table, then, is aligned with a similar set of coordinate axes, the x′-axis running the width of the table, the positive x′ direction toward the right of the table as shown in the photograph, and the y′-axis running across the depth of the table, the positive y′ direction toward the front of the table as shown in the photograph. If the table is aligned with the lecture hall so that it faces the class as it is shown in the photograph, then the two sets of coordinate axes are aligned with each other, and the positive x′ direction is the same as the positive x direction, and the positive y′ direction is the same as the postive y direction.

The x and y components of the velocity of the table relative to the lecture hall, then, are v_x and v_y, those of the cart relative to the table are v_x′ and v_y′ (= 0), and the components of the velocity of the cart relative to the lecture hall are v_x + v_x′ and v_y + 0.

If the table is stationary, then if you move the cart along the track, it has a velocity in the x′ direction, v_x′, but none in the y′ direction. Since v_x and v_y both equal zero, the velocity of the cart relative to the lecture hall is also v_x′.

If you move the table sideways, but give the cart an equal velocity in the opposite direction with respect to the table, then v_x′ = -v_x, and since v_y = 0, the velocity of the table with respect to the lecture hall is v_x, and the velocity of the cart with respect to the lecture hall equals zero. (This might be tricky, but if you could hold the cart as you move the table, or have an assistant help you do this, it should be possible.)

If you move the table sideways, but give the cart zero velocity with respect to the table (and it stays in the same place with respect to the table; the same note as above applies regarding how to accomplish this), then the velocities of the both the table and the cart with respect to the lecture hall equal v_x.

If you give the cart some velocity with respect to the table and then move the table sideways, with respect to the lecture hall the velocity of the table is v_x, and the velocity of the cart is v_x + v_x′.

If the cart is stationary with respect to the table and you move the table toward the class, then both v_x and v_x′ equal zero, and with respect to the lecture hall, the velocity of both the table and the cart is v_y.

If you give the cart some velocity with respect to the table and then move the table toward the class, then with respect to the lecture hall, the velocity of the table is v_y′, and the velocity of the cart is v_x′ + v_y. The motion of the table is in one dimension, but the motion of the cart is now two-dimensional.

You can also move the table in both the x and y directions simultaneously, or rotate it, to show whatever kind of relative motion you wish.