91Ó°ÊÓ

The velocity function is fundamentally important in describing how an object moves. For an electron moving in space, its position as a function of time is denoted by a vector. In this problem, the position vector given is \( \vec{r}(t) = 3.00t \hat{\mathrm{i}} - 4.00t^2 \hat{\mathrm{j}} + 2.00 \hat{\mathrm{k}} \). To find out how fast and in what direction the electron is moving at any time \( t \), we compute the velocity function.

• The velocity \( \vec{v}(t) \) is the derivative of the position vector with respect to time \( t \).


• We differentiate each component separately: the \( \hat{\mathrm{i}} \) component is the derivative of \( 3.00t \), the \( \hat{\mathrm{j}} \) component is the derivative of \( -4.00t^2 \), and the \( \hat{\mathrm{k}} \) component is constant, so its derivative is zero.

So, the calculated velocity function is \( \vec{v}(t) = 3.00 \hat{\mathrm{i}} - 8.00t \hat{\mathrm{j}} \). Here, - \( 3.00 \hat{\mathrm{i}} \) represents constant motion in the x-direction, and - \( -8.00t \hat{\mathrm{j}} \) indicates that the motion in the y-direction changes over time.

Knowing the magnitude of a velocity vector gives you the speed of the object. Speed is the absolute value of how fast something is moving, regardless of its direction. Let's explore how we determine this magnitude from the velocity vector \( \vec{v} = 3.00 \hat{\mathrm{i}} - 24.00 \hat{\mathrm{j}} \) at \( t = 3.00 \) seconds.

• The magnitude of the velocity vector can be calculated using the Pythagorean theorem.


• It involves taking the square root of the sum of the squares of its components: \( \sqrt{(3.00)^2 + (-24.00)^2} \).


• This calculation simplifies to \( \sqrt{9 + 576} = \sqrt{585} \approx 24.19 \text{ m/s} \).

This magnitude tells us how fast the electron is moving at exactly \( t = 3.00 \) seconds, without any consideration of which direction it is heading. Speed is useful in various practical situations where only how much motion exists is important, not the direction.

Determining the angle of the velocity vector helps us understand the direction of the electron's movement in relation to the axes. In this particular problem, we calculate the angle \( \theta \) relative to the positive x-axis.

• We use the inverse tangent function to compute this angle, taking the ratio of the y-component to the x-component: \( \theta = \tan^{-1}\left(\frac{-24.00}{3.00}\right) \).


• Mathematically, this gives \( \theta \approx \tan^{-1}(-8.00) \), which evaluates to approximately \( -82.87^\circ \).


• The negative sign indicates that the direction is clockwise relative to the positive x-axis, suggesting the electron is moving downwards and towards the left.

Angle calculation is crucial for visualizing the trajectory of the electron and can help in comprehending complex movement patterns in multidimensional spaces.