Question

Relationship between \(\mathbf{T}, \mathbf{N},\) and a Show that if an object accelerates in the sense that \(\frac{d^{2} s}{d t^{2}}>0\) and \(\kappa \neq 0,\) then the acceleration vector lies between \(\mathbf{T}\) and \(\mathbf{N}\) in the plane of \(\mathbf{T}\) and \(\mathbf{N}\). Show that if an object decelerates in the sense that \(\frac{d^{2} s}{d t^{2}}<0,\) then the acceleration vector lies in the plane of \(\mathbf{T}\) and \(\mathbf{N},\) but not between \(\mathbf{T}\) and \(\mathbf{N} .\)

Short answer

In summary, if an object has positive acceleration (\(\frac{d^2s}{dt^2} > 0\)) and non-zero curvature (\(\kappa \neq 0\)), its acceleration vector lies between the tangent vector and the normal vector in the plane of T and N. On the other hand, if an object decelerates (\(\frac{d^2s}{dt^2} < 0\)), its acceleration vector lies in the plane of T and N but not between T and N.

Step-by-step solution

Expressing Acceleration Vector in terms of Tangent Vector and Normal Vector

: The acceleration vector, \(\mathbf{a}\), can be written as the sum of the tangential component of acceleration, a_T, and the normal component of acceleration, a_N: \(\mathbf{a} = a_{T} \mathbf{T} + a_{N} \mathbf{N}\) where T is the unit tangent vector and N is the unit normal vector. To prove that, we need to find expressions for \(a_T\) and \(a_N\) in terms of \(ds\) and \(d^2s\).

Finding the expression for \(a_T\)

: Notice that \(a_T = \frac{d^2s}{dt^2}\), so we have: \(a_{T} = \frac{d^2s}{dt^2}\)

Finding the expression for \(a_N\)

: \(a_N\) can be derived from the relation \(a_N = \kappa (\frac{ds}{dt})^2\), where \(\kappa\) is the curvature. Then: \(a_{N} = \kappa (\frac{ds}{dt})^2\)

Analyzing the relationship for objects with positive acceleration and non-zero curvature

: Given that \(\frac{d^2s}{dt^2} > 0\), and since \(\kappa \neq 0\), then both \(a_T > 0\) and \(a_N > 0\). Using the expressions for \(a_{T}\) and \(a_{N}\) derived in Step 2 and Step 3, we can rewrite the acceleration vector as: \(\mathbf{a} = a_{T} \mathbf{T} + a_{N} \mathbf{N} = (\frac{d^2s}{dt^2})\mathbf{T} + (\kappa(\frac{ds}{dt})^2)\mathbf{N}\) Since \(a_T > 0\) and \(a_N > 0\), the acceleration vector \(\mathbf{a}\) lies between the tangent vector \(\mathbf{T}\) and the normal vector \(\mathbf{N}\) in the plane of T and N.

Analyzing the relationship for objects with negative acceleration

: Given that \(\frac{d^2s}{dt^2} < 0\), it follows that \(a_T < 0\). The curvature, \(\kappa\), is always non-negative (considering the case of a decelerating object \(\kappa \neq 0\)). Thus, \(a_N \geq 0\). Using the expressions for \(a_{T}\) and \(a_{N}\) derived in Step 2 and Step 3, we can rewrite the acceleration vector as: \(\mathbf{a} = a_{T} \mathbf{T} + a_{N} \mathbf{N} = (\frac{d^2s}{dt^2})\mathbf{T} + (\kappa(\frac{ds}{dt})^2)\mathbf{N}\) Since \(a_T < 0\) and \(a_N \geq 0\), the acceleration vector \(\mathbf{a}\) lies in the plane of T and N, but not between \(\mathbf{T}\) and \(\mathbf{N}\).

Key concepts

Acceleration Vector

In the realm of calculus and vectors, understanding the acceleration vector is crucial for analyzing how an object's motion changes over time. Generally, we express the acceleration vector, \( \mathbf{a} \), in terms of two components: the tangential component, \( a_T \), and the normal component, \( a_N \). The formula is given by:

• \( \mathbf{a} = a_{T} \mathbf{T} + a_{N} \mathbf{N} \)

Here, \( \mathbf{T} \) and \( \mathbf{N} \) represent the tangent and normal unit vectors, respectively. The tangential component, \( a_T \), is responsible for the change in the speed of the object along its path. Meanwhile, the normal component, \( a_N \), is linked to how sharply an object turns along a curved trajectory.
Since \( a_T \) can be found using the second derivative of speed, \( a_{T} = \frac{d^2s}{dt^2} \), this implies that if an object is accelerating, \( \frac{d^2s}{dt^2} > 0 \), then \( a_T > 0 \). The normal component, \( a_N = \kappa (\frac{ds}{dt})^2 \), reveals that with non-zero curvature \( \kappa eq 0 \), \( a_N \) contributes to the object's ability to navigate curves.
Understanding these components lets us predict not just where an object will go, but how its speed and direction are evolving over time.

Tangent Vector

The tangent vector \( \mathbf{T} \) plays a significant role in identifying the immediate direction of an object in motion. It is a unit vector that always points in the direction of the path an object is taking and is tangent to the curve at any point. This means the direction of \( \mathbf{T} \) is directly aligned with the velocity vector of the object as it moves.
Since \( \mathbf{T} \) is a unit vector, it maintains a constant length of 1, emphasizing that its primary purpose is directional rather than magnitude-based descriptions of movement. Whenever we discuss the acceleration vector being aligned with the tangent vector, we're focusing on the part of acceleration that affects speed. If a car on a racing track speeds up or slows down along a straight section, this reflects changes in the tangential component \( a_T \), aligning with \( \mathbf{T} \).
The tangent vector’s role helps us understand geometric properties of motion on curved paths. Analyzing how \( \mathbf{T} \) interacts with other vectors, such as the normal vector, builds up a full picture of the dynamics and trajectories during any motion, be it linear or curvilinear.

Normal Vector

The normal vector, represented as \( \mathbf{N} \), is another useful vector in calculus that complements the tangent vector in describing an object's motion through space. Unlike \( \mathbf{T} \), which denotes the path's immediate direction, \( \mathbf{N} \) is perpendicular to \( \mathbf{T} \) and points toward the direction an object will turn as it follows a curved path.
\( \mathbf{N} \) is particularly crucial in situations where curvature affects motion. If an object is following a sharply bending path, \( \mathbf{N} \) is what determines how drastically the path turns, being directly proportional to the curvature \( \kappa \). Thus, when analyzing the acceleration vector, \( a_N \) essentially denotes the strength of this turning effect.

• \( a_N = \kappa (\frac{ds}{dt})^2 \)

Here, \( a_N \) doesn’t just capture the object’s speed but also how sharply it’s moving along a curve, making the normal vector key to examining centripetal motion. Whether an object is speeding up or slowing down, its handling through these turns is largely seen through \( \mathbf{N} \), emphasizing its importance in vector calculus.