Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Understanding Tangential and Normal Components of Acceleration
In this exercise, we aim to find the tangential and normal components of acceleration at a specified time \( t \) for a given plane curve \( \mathbf{r}(t) \).
#### Problem Statement:
Given that the position vector as a function of time \( t \) is:
\[ \mathbf{r}(t) = t \mathbf{i} + \frac{5}{t} \mathbf{j}, \]
find the tangential and normal components of acceleration when \( t = 2 \).
#### Components of Acceleration:
- **Tangential acceleration (\( a_T \))**: This represents the rate of change of speed (the magnitude of velocity).
- **Normal acceleration (\( a_N \))**: This represents the rate of change of direction of the velocity.
#### Required calculations:
To solve this problem, we typically follow these steps:
1. Compute the first derivative of \( \mathbf{r}(t) \) to find the velocity \( \mathbf{v}(t) \).
2. Compute the second derivative of \( \mathbf{r}(t) \) to determine the acceleration \( \mathbf{a}(t) \).
3. Find the magnitudes of velocity and acceleration.
4. Use the formulas for \( a_T \) and \( a_N \) based on these magnitudes.
#### Placeholder for Solutions:
\[ a_T = \boxed{} \]
\[ a_N = \boxed{} \]
Feedback shows that the initial attempts to find these components were incorrect (indicated by the red "X" marks). Re-evaluate the derivatives and calculations to determine the correct components of the acceleration.
Understanding how these components work will give you a profound insight into the kinematics of motion along a path, enriching your grasp of fundamental concepts in classical mechanics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe18dfc87-3c35-485c-a41e-ee4c781c7cac%2F716d0a79-553b-4c93-ba0c-2f24fcca11d7%2Fxwxczx_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Tangential and Normal Components of Acceleration
In this exercise, we aim to find the tangential and normal components of acceleration at a specified time \( t \) for a given plane curve \( \mathbf{r}(t) \).
#### Problem Statement:
Given that the position vector as a function of time \( t \) is:
\[ \mathbf{r}(t) = t \mathbf{i} + \frac{5}{t} \mathbf{j}, \]
find the tangential and normal components of acceleration when \( t = 2 \).
#### Components of Acceleration:
- **Tangential acceleration (\( a_T \))**: This represents the rate of change of speed (the magnitude of velocity).
- **Normal acceleration (\( a_N \))**: This represents the rate of change of direction of the velocity.
#### Required calculations:
To solve this problem, we typically follow these steps:
1. Compute the first derivative of \( \mathbf{r}(t) \) to find the velocity \( \mathbf{v}(t) \).
2. Compute the second derivative of \( \mathbf{r}(t) \) to determine the acceleration \( \mathbf{a}(t) \).
3. Find the magnitudes of velocity and acceleration.
4. Use the formulas for \( a_T \) and \( a_N \) based on these magnitudes.
#### Placeholder for Solutions:
\[ a_T = \boxed{} \]
\[ a_N = \boxed{} \]
Feedback shows that the initial attempts to find these components were incorrect (indicated by the red "X" marks). Re-evaluate the derivatives and calculations to determine the correct components of the acceleration.
Understanding how these components work will give you a profound insight into the kinematics of motion along a path, enriching your grasp of fundamental concepts in classical mechanics.
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