What are (a) the average velocity and (b) the average acceleration of the tip of the 2.4 -cm-long hour hand of a clock in the interval from noon to 6 PM? Use unit vector notation, with the \(x\) -axis pointing toward 3 and the \(y\) -axis toward noon.

Understanding unit vector notation is crucial when dealing with vector quantities in physics. A unit vector is a vector that has a magnitude of exactly one and points in a particular direction. It's used to represent directional information and is often denoted with a 'hat' symbol, such as \(\hat{i}\) or \(\hat{j}\). In a two-dimensional coordinate system, \(\hat{i}\) and \(\hat{j}\) specifically refer to the unit vectors pointing in the directions of the positive x-axis and y-axis, respectively.

For example, in the clock problem, the hour hand's position at noon pointing towards 6 is given by the vector \( -\hat{j} \), indicating it points in the negative direction of the y-axis. Similarly, by 6 PM, as it points toward 9, it's given by the vector \( -\hat{i} \), now pointing in the negative x-axis direction. The change in these directions, described using unit vectors, lays the foundation for vector displacement and velocity calculations.

Displacement is a vector quantity that represents a change in position. To calculate displacement, we identify the initial and final positions of an object and determine the straight-line path connecting these two points. The magnitude of this path, along with the direction, gives us the displacement vector.

In the context of our clock example, the hour hand's initial and final positions create an isosceles right triangle with the clock's center as the vertex. Using Pythagorean theorem, we find the hypotenuse, which represents the magnitude of displacement. It's essential to consider both the magnitude and direction to express the displacement in unit vector notation fully, which was shown by converting the calculated length into meters and then using unit vectors to denote the new positions of the hour hand at noon and 6 PM. This comprehensive approach captures the actual 'movement' of the hour hand across the clock's face.

Average acceleration is the rate of change of velocity over a given time interval. It's another vector quantity, typically measured in meters per second squared \(m/s^2\). Average acceleration is given by the change in velocity divided by the time during which this change occurs.

For the hour hand of a clock, which moves at a constant speed in a circular path, its velocity vector changes direction, but its speed (magnitude of the velocity) remains constant. Thus, when dealing with such rotational motion in a time interval where there's no change in speed, the average acceleration is zero because the change in velocity is zero. This concept can be tricky since there is movement and positional change, but in terms of velocity - speed and direction - there's no 'acceleration' in the traditional sense. It reinforces the idea that acceleration is not just about speed changes, but also directional changes.