91Ó°ÊÓ

The concept of a velocity vector is essential to understanding motion in a three-dimensional space. A velocity vector describes the speed and direction of an object. It is derived from the position vector by calculating its time derivative. In the given problem, the position vector is \( \vec{r}=(3500-160 t) \hat{\mathrm{i}}+2700 \hat{\mathrm{j}}+300 \hat{\mathrm{k}} \), and so, to find the velocity vector \( \vec{v} \), we differentiate with respect to time \( t \). This gives us \( \vec{v} = \frac{d\vec{r}}{dt} = (-160) \hat{\mathrm{i}} \). The vector indicates:

• a speed of 160 meters per second.
• a direction along the negative \( x \)-axis, meaning toward the west.

This singular direction and speed signify uniform or constant motion along a straight path without any deviation into the \( y \) or \( z \) axes.

Direction of motion is straightforward when you have a clear velocity vector. Here, the velocity vector \( \vec{v} = -160 \hat{\mathrm{i}} \) exemplifies this concept by pointing towards a defined direction. Since the velocity has a negative component only in the \( \hat{\mathrm{i}} \) direction, the motion is purely westward in alignment with the negative \( x \)-axis. This makes it convenient to understand where the object is heading:

• If an axis is positive, moving in the negative direction means opposite to that sense.
• The absence of components in \( \hat{\mathrm{j}} \) or \( \hat{\mathrm{k}} \) shows no movements north or vertically up, respectively.

Thus, an understanding of vectors straight away tells you all about the directional movement of objects in a physics problem.

In physics, net force propels an object, but understanding how it plays with constant velocity helps solve problems like this. Newton's second law \( \vec{F} = m \cdot \vec{a} \) relates force, mass, and acceleration. Here, the velocity is constant, \( \vec{v} = -160 \hat{\mathrm{i}} \), indicating no change in speed or direction. Therefore, acceleration \( \vec{a} \) is zero. With zero acceleration:

• The net force \( \vec{F} = 250 \times 0 = 0 \) Newtons.
• This indicates a state of equilibrium where all external forces, if any, balance out, resulting in steadiness in motion.

Knowing there's no net force helps understand that steady motion signifies balance without additional push or pull.

Acceleration tells us about the changes in velocity. A non-zero acceleration implies changes in speed, direction, or both. In our problem, we already know the velocity \( \vec{v} \) is constant (-160 meters per second in the \( \hat{\mathrm{i}} \) direction). To find acceleration, you need to differentiate velocity with time:

• If \( \vec{v} \) is constant: \( \vec{a} = \frac{d\vec{v}}{dt} = 0 \).

The zero acceleration means:

• The object maintains a steady speed.
• The direction of travel does not alter.

An understanding of acceleration in such problems clarifies that an object in such a state continues moving uniformly without any unexpected changes.