Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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### Problem Statement Given the vector function \(\vec{r}(t) = \langle t, e^t, 10 \rangle\), find the tangential component of acceleration \(a_T\) when \(t = 7\). --- ### Solution To find the tangential component of acceleration \(a_T\), we follow these steps: 1. **Compute the velocity vector \(\vec{v}(t)\):** \[ \vec{v}(t) = \frac{d\vec{r}}{dt} = \left\langle \frac{d}{dt}(t), \frac{d}{dt}(e^t), \frac{d}{dt}(10) \right\rangle = \langle 1, e^t, 0 \rangle \] 2. **Compute the acceleration vector \(\vec{a}(t)\):** \[ \vec{a}(t) = \frac{d\vec{v}}{dt} = \left\langle \frac{d}{dt}(1), \frac{d}{dt}(e^t), \frac{d}{dt}(0) \right\rangle = \langle 0, e^t, 0 \rangle \] 3. **Find the magnitude of the velocity vector \(|\vec{v}(t)|\):** \[ |\vec{v}(t)| = \sqrt{ (1)^2 + (e^t)^2 + (0)^2 } = \sqrt{1 + e^{2t}} \] 4. **Calculate the tangential component of acceleration \(a_T\):** \[ a_T = \frac{\vec{v}(t) \cdot \vec{a}(t)}{|\vec{v}(t)|} \] Here, \(\vec{v}(t) \cdot \vec{a}(t)\) is the dot product of the velocity and acceleration vectors: \[ \vec{v}(t) \cdot \vec{a}(t) = \langle 1, e^t, 0 \rangle \cdot \langle 0, e^t, 0 \rangle = 1 \cdot 0 + e^t \cdot
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Transcribed Image Text:### Problem Statement Given the vector function \(\vec{r}(t) = \langle t, e^t, 10 \rangle\), find the tangential component of acceleration \(a_T\) when \(t = 7\). --- ### Solution To find the tangential component of acceleration \(a_T\), we follow these steps: 1. **Compute the velocity vector \(\vec{v}(t)\):** \[ \vec{v}(t) = \frac{d\vec{r}}{dt} = \left\langle \frac{d}{dt}(t), \frac{d}{dt}(e^t), \frac{d}{dt}(10) \right\rangle = \langle 1, e^t, 0 \rangle \] 2. **Compute the acceleration vector \(\vec{a}(t)\):** \[ \vec{a}(t) = \frac{d\vec{v}}{dt} = \left\langle \frac{d}{dt}(1), \frac{d}{dt}(e^t), \frac{d}{dt}(0) \right\rangle = \langle 0, e^t, 0 \rangle \] 3. **Find the magnitude of the velocity vector \(|\vec{v}(t)|\):** \[ |\vec{v}(t)| = \sqrt{ (1)^2 + (e^t)^2 + (0)^2 } = \sqrt{1 + e^{2t}} \] 4. **Calculate the tangential component of acceleration \(a_T\):** \[ a_T = \frac{\vec{v}(t) \cdot \vec{a}(t)}{|\vec{v}(t)|} \] Here, \(\vec{v}(t) \cdot \vec{a}(t)\) is the dot product of the velocity and acceleration vectors: \[ \vec{v}(t) \cdot \vec{a}(t) = \langle 1, e^t, 0 \rangle \cdot \langle 0, e^t, 0 \rangle = 1 \cdot 0 + e^t \cdot
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