2D Motion (part1)

Main Ideas

1. Six Quantities Used to Describe Motion

Motion in two dimensions uses the same six basic quantities as motion in one dimension:

• Position (\mathbf r): a vector from a chosen origin to the object’s location.
• Displacement (\Delta\mathbf r): the change in position, (\mathbf r_f-\mathbf r_i).
• Velocity (\mathbf v): a vector describing how position changes with time.
• Acceleration (\mathbf a): a vector describing how velocity changes with time.
• Distance: the total length of the path travelled; a scalar.
• Speed: the magnitude of velocity; a scalar.

Position, displacement, velocity, and acceleration are vectors and can be split into (x)- and (y)-components. Distance and speed have magnitude only and are nonnegative.

2. Position, Displacement, and Distance Are Different

The origin is chosen by the observer. Once it is chosen, a location can be described by its (x)- and (y)-coordinates. A position vector points directly from the origin to the object.

Displacement connects the initial and final positions:

[ \Delta\mathbf r=\mathbf r_f-\mathbf r_i =(x_f-x_i)\mathbf i+(y_f-y_i)\mathbf j. ]

It is the straight-line change in position and does not depend on the route taken. Distance, by contrast, is the length of the actual path. It can differ for different routes between the same endpoints and is at least as large as the magnitude of displacement.

For a curved path, its length can be estimated by adding the lengths of many short straight segments along it. Making the segments increasingly small leads to the idea of integration. The lesson notes that this calculation is not needed for the displacement examples.

3. Position and Path Illustrations

The instructor uses position functions to show how an object’s location changes over time. For example, with (x=t^2+5t) and (y=7t), the initial position is the origin, and at (t=1) the position is (6\mathbf i+7\mathbf j). Plotting (x(t)) and (y(t)) together produces a parametric path.

The examples distinguish the object’s curved route from its straight-line displacement between two times. An online parametric plotting tool is used to visualize such paths.

4. Velocity Is Tangent to the Path

Velocity is the time derivative of position:

[ \mathbf v=\frac{d\mathbf r}{dt} =\frac{dx}{dt}\mathbf i+\frac{dy}{dt}\mathbf j. ]

The velocity vector points in the direction the object is moving at that instant. For a curved path, it is tangent to the path. The instructor illustrates this with a path for which the (x)-component of velocity grows over time while the (y)-component stays constant; consequently, the velocity vector turns toward the horizontal.

5. Acceleration Changes Velocity

Acceleration is the time derivative of velocity:

[ \mathbf a=\frac{d\mathbf v}{dt}. ]

In two dimensions, acceleration can:

• Change the magnitude of velocity (the speed).
• Change its direction.
• Change both.

Thus, an object can have nonzero acceleration even when its speed stays constant—for example, when it follows a circular path and its velocity continually changes direction. If neither the magnitude nor direction of velocity changes, acceleration is zero.

The instructor contrasts this with one-dimensional motion, where velocity changes along a single line: acceleration may increase or decrease the velocity’s magnitude, with its sign indicating the direction of the change.

6. Average Velocity and Average Acceleration

The definitions emphasized are:

[ \mathbf v_{\text{avg}}=\frac{\Delta\mathbf r}{\Delta t}, \qquad \mathbf a_{\text{avg}}=\frac{\Delta\mathbf v}{\Delta t}. ]

The lesson cautions against assuming that the average of the endpoint velocities always gives average velocity. The reliable method is to calculate the change in position over the elapsed time. Averaging endpoint values works in particular cases, such as when the relevant quantity changes linearly over the interval.

7. Piecewise-Motion Example

The instructor considers a particle starting with velocity (2\mathbf i-3\mathbf j), with no acceleration for the first 3 seconds, acceleration (\mathbf i+\mathbf j) from (t=3) to (t=5), and no acceleration afterward. Taking the initial position as the origin:

• From (t=0) to (t=3), the position changes by (6\mathbf i-9\mathbf j).
• At (t=4), the velocity is (3\mathbf i-2\mathbf j).
• At (t=5), the velocity is (4\mathbf i-\mathbf j), and the position is (12\mathbf i-13\mathbf j).
• At (t=6), after one further second at constant velocity, the position is (16\mathbf i-14\mathbf j).
• Therefore, the displacement over the first 6 seconds is (16\mathbf i-14\mathbf j), assuming the initial position was the origin.

These values follow from the stated initial velocity and acceleration. Some intermediate arithmetic in the auto-generated subtitles is inconsistent with those stated inputs.

8. Applications Introduced

Projectile motion and circular motion are named as important two-dimensional cases to be studied. The discussion of circular motion reinforces that acceleration can continually turn the velocity vector, even if speed does not change.

Speakers and Sources Featured

• Speaker: One physics instructor or lecturer; no name is given in the subtitles.
• Tool mentioned: An unnamed online parametric-equation plotting tool.
• No other identifiable speakers or sources are featured.