91Ó°ÊÓ

In physics, velocity is a vector quantity, which means it has both magnitude and direction. It describes how fast an object is moving and in which direction. In simpler terms, while speed tells us how quickly something is happening, velocity tells us how quickly and in which direction it is happening.

When we talk about velocity in the context of motion in three-dimensional space, it can be represented in unit-vector notation. This is a way of expressing velocity using the standard basis vectors: \( \hat{\mathrm{i}} \), \( \hat{\mathrm{j}} \), and \( \hat{\mathrm{k}} \). These vectors point in the directions of the \( x \)-axis, \( y \)-axis, and \( z \)-axis respectively.

• The \( \hat{\mathrm{i}} \) component measures velocity along the \( x \)-axis.
• The \( \hat{\mathrm{j}} \) component measures velocity along the \( y \)-axis.
• The \( \hat{\mathrm{k}} \) component measures velocity along the \( z \)-axis.

Together, these components describe the complete motion of an object, such as an electron in our problem. Here, the electron’s velocity at any time \( t \) is given by \( \vec{v}(t) = 3.00 \hat{\mathrm{i}} - 8.00 t \hat{\mathrm{j}} \). This tells us how fast and in which directions the electron is moving at any given moment.

Position is another fundamental concept in kinematics. It refers to an object's specific location in space at a given time. In our exercise, the electron's position is described by the vector equation \( \vec{r}=3.00 t \hat{\mathrm{i}} - 4.00 t^{2} \hat{\mathrm{j}} + 2.00 \hat{\mathrm{k}} \), where each term indicates the electron’s location along the \( x \), \( y \), and \( z \) axes respectively.

This expression allows you to determine where the electron is at any time \( t \). At \( t=0 \), for instance, the electron is at the origin shifted slightly on the \( z \) axis due to the \( 2.00 \hat{\mathrm{k}} \) component. It’s essential to have a clear understanding of position as it forms the basis for determining other aspects like velocity and acceleration.

• The \( \hat{\mathrm{i}} \) component \( 3.00 t \) shows the position changes linearly with time along the \( x \)-axis.
• The \( \hat{\mathrm{j}} \) component \( -4.00 t^2 \) shows a quadratic dependence on time along the \( y \)-axis, indicating a more complex motion.
• The \( \hat{\mathrm{k}} \) component remains constant, revealing no motion along the \( z \)-axis over time.

Understanding position aids in deriving velocity by taking its derivative with respect to time.

Unit-vector notation is an incredibly useful way to express vectors, especially in physics. This notation breaks a vector down into its components, which makes calculations easier and reveals how each part of the vector relates to physical space.

Unit vectors, \( \hat{\mathrm{i}}, \hat{\mathrm{j}}, \hat{\mathrm{k}} \), have a magnitude of one and point along the positive \( x \), \( y \), and \( z \) axes respectively. Any vector can be written as a sum of these, scaled by how much the vector 'points' in each direction. In our exercise for example, the electron's position is written as: \( \vec{r} = 3.00 t \hat{\mathrm{i}} - 4.00 t^2 \hat{\mathrm{j}} + 2.00 \hat{\mathrm{k}} \).

These components tell us:

• For \( \hat{\mathrm{i}} \), \( 3.00 t \) indicates how the position changes along the \( x \)-direction over time.
• For \( \hat{\mathrm{j}} \), \( -4.00 t^2 \) indicates the change along the \( y \)-direction, having a quadratic nature.
• \( \hat{\mathrm{k}} \) shows constant position over time along the \( z \)-direction.

Expressing vectors in unit-vector notation provides a clear, concise view of how an object's position or velocity changes across each axis of motion.

Differentiation is a vital tool in calculus, particularly in analyzing motion in kinematics. It helps us understand how one quantity changes with respect to another. When applied to position, it allows us to derive velocity, which is essentially the rate of change of position with time.

In the context of our problem, the electron's position function is given by \( \vec{r}(t) = 3.00 t \hat{\mathrm{i}} - 4.00 t^2 \hat{\mathrm{j}} + 2.00 \hat{\mathrm{k}} \). To find the velocity which tells us how the electron’s position changes with time, we differentiate each component separately with respect to time \( t \).

The process goes like this:

• Differentiating \( 3.00 t \hat{\mathrm{i}} \) gives \( 3.00 \hat{\mathrm{i}} \), indicating a constant velocity component along the \( x \)-axis.
• Differentiating \( -4.00 t^2 \hat{\mathrm{j}} \) gives \( -8.00 t \hat{\mathrm{j}} \), showing a velocity component along the \( y \)-axis that changes with time.
• The \( 2.00 \hat{\mathrm{k}} \) part remains \( 0 \hat{\mathrm{k}} \) because differentiating a constant yields zero, which means no velocity component along the \( z \)-axis.

Thus, differentiation transforms our position vector into a velocity vector: \( \vec{v}(t) = 3.00 \hat{\mathrm{i}} - 8.00 t \hat{\mathrm{j}} \). Understanding differentiation allows us to explore motion in a detailed and precise manner.