Question

Use a computer algebra system to graph the space curve. Then find \(\mathbf{T}(t), \mathbf{N}(t), a_{\mathrm{T}},\) and \(a_{\mathrm{N}}\) at the given time \(t\). Sketch \(\mathbf{T}(t)\) and \(\mathbf{N}(t)\) on the space curve. $$ \mathbf{r}(t)=t \mathbf{i}+3 t^{2} \mathbf{j}+\frac{t^{2}}{2} \mathbf{k} \quad t=2 $$

Short answer

First graph the vector function to have a clear visual representation. Compute the unit tangential vector \( \mathbf{T}(t) \) and the unit normal vector \( \mathbf{N}(t) \) using their respective formulas. Finally find the tangential acceleration \( a_{T} \) and the normal acceleration \( a_{N} \) using their respective formulas. All these calculations are evaluated at \( t=2 \). Finally sketch the vectors \( \mathbf{T}(t) \) and \( \mathbf{N}(t) \) on the space curve.

Step-by-step solution

Graph the Space Curve

To graph the space curve defined by the vector \( \mathbf{r}(t) = t \mathbf{i} + 3t^2 \mathbf{j} + \frac{t^2}{2} \mathbf{k}\), input the function into a computer algebra system and generate the plot.

Computing \( \mathbf{T}(t) \)

To find the unit tangential vector \( \mathbf{T}(t) \), first compute the derivative of \( \mathbf{r}(t) \). This produces \( \mathbf{r}'(t) = \mathbf{i} + 6t \mathbf{j} + t \mathbf{k}\). Then, find the magnitude of this vector as \( ||\mathbf{r}'(t)|| = \sqrt{1^2 + (6t)^2 + t^2}\) which simplifies to \( ||\mathbf{r}'(t)|| = \sqrt{1 + 36t^2 + t^2}\). This is then inserted into the formula \( \mathbf{T}(t) = \frac{\mathbf{r}'(t)}{||\mathbf{r}'(t)||}\) to find \( \mathbf{T}(t) \) at \( t=2 \).

Compute \( \mathbf{N}(t) \)

To find the unit normal vector \( \mathbf{N}(t) \), we first need to compute the derivative of \( \mathbf{T}(t) \), called \( \mathbf{T}'(t) \). Then, we find the magnitude \( ||\mathbf{T}'(t)|| \) and use it in the formula \( \mathbf{N}(t) = \frac{\mathbf{T}'(t)}{||\mathbf{T}'(t)||} \) to find \( \mathbf{N}(t) \) at \( t=2 \).

Compute \( a_{T} \) and \( a_{N} \)

The tangential acceleration \( a_{T} \) can be found using the formula \( a_{T} = \mathbf{V}'(t) . \mathbf{T}(t) \) whereas the normal acceleration \( a_{N} \) is found using \( a_{N} = \mathbf{V}'(t) . \mathbf{N}(t) \) where \( \mathbf{V}'(t) \) is the second derivative of vector \( \mathbf{r}(t) \). Evaluate these formulas at \( t=2 \).

Sketch \( \mathbf{T}(t) \) and \( \mathbf{N}(t) \) on the Space Curve

You have found the magnitudes and directions of the vectors \( \mathbf{T}(t) \) and \( \mathbf{N}(t) \) at \( t=2 \) in step 2 and step 3 respectively. Using these values, sketch these vectors by originating them at the point on the curve where \( t=2 \). Make sure the lengths of the vectors represent their magnitudes and their directions are towards increasing \( t \).

Key concepts

Vector Calculus

Vector calculus is a cornerstone of mathematics that deals with the differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space. Primarily, it provides us with tools to analyze and describe the motion and behavior of objects moving along curves in space.

In vector calculus, the graphical representation of vector functions plays a pivotal role. For instance, in the exercise given, \(\mathbf{r}(t) = t \mathbf{i} + 3t^2 \mathbf{j} + \frac{t^2}{2} \mathbf{k}\) is a vector function that gives the position of a particle in space at any time \(t\). The curve formed by this vector function as \(t\) varies is called a space curve.

To visualize the path of an object, graphing the space curve using a computer algebra system can help. The curve's direction at any point can be represented by the tangent vector, and changes to the curve's direction can be examined using the concept of curvature and normal vectors. Understanding these vector functions and their derivatives is fundamental to unleashing the full potential of vector calculus.

Tangential and Normal Vectors

In the study of motion along a space curve, tangential and normal vectors—often denoted as \(\mathbf{T}(t)\) and \(\mathbf{N}(t)\), respectively—serve to unravel the direction of travel and the instantaneous change of direction at a given point along the curve.

Firstly, the unit tangent vector \(\mathbf{T}(t)\) is obtained by differentiating the position vector \(\mathbf{r}(t)\) and normalizing it. This vector always points in the direction of motion, and its length is equal to 1, ensuring that it is indeed a unit vector. The process involves taking the derivative \(\mathbf{r}'(t)\) which gives us the velocity, then normalizing it to find \(\mathbf{T}(t)\).

The unit normal vector \(\mathbf{N}(t)\), on the other hand, is perpendicular to the unit tangent vector, pointing towards the center of the path's curvature. It explains how the path is changing direction at any point in time. To find it, one takes the derivative of the unit tangent vector to get \(\mathbf{T}'(t)\) and then once again normalize it.

Together, these vectors provide a comprehensive picture of an object’s motion along a curve, helping mathematicians and physicists to easily describe the velocity and acceleration of the moving object.

Tangential and Normal Accelerations

When an object moves along a space curve, not only is it important to understand its velocity, but also how that velocity is changing—that is, its acceleration. Acceleration can be decomposed into two perpendicular components: tangential acceleration \(a_{\mathrm{T}}\) and normal acceleration \(a_{\mathrm{N}}\).

Tangential acceleration \(a_{\mathrm{T}}\) is the rate of change of the object’s speed along the curve and is tangent to the path. It can be found by taking the derivative of the velocity vector (which is the same as the second derivative of the position vector \(\mathbf{r}(t)\)) and projecting this acceleration vector onto the unit tangent vector \(\mathbf{T}(t)\). This component shows how quickly the object is speeding up or slowing down along the curve.

Normal acceleration, \(a_{\mathrm{N}}\), is oriented towards the curve's center of curvature and represents how fast the direction of the velocity is changing. This is quantified by projecting the acceleration vector onto the unit normal vector \(\mathbf{N}(t)\). \(a_{\mathrm{N}}\) is also related to the curve's curvature and the object’s speed.

These concepts are vital for understanding complex motions, like those of a roller coaster or a planet orbiting a star. Without separating acceleration into tangential and normal components, it would be challenging to thoroughly analyze the forces at work on an object traveling along a curve in space.