calculus, section 13.4 - motion in space

Find the tangential and normal components of the acceleration vector

r(t) = t i + 2et j + e2t k

aT	=	
  
aN	=

To find the tangential (\( a_T \)) and normal (\( a_N \)) components of the acceleration vector, we follow these steps:

Given the position vector \( \mathbf{r}(t) = t \hat{i} + 2e^t \hat{j} + e^{2t} \hat{k} \), we need to:

1. Differentiate the position vector \( \mathbf{r}(t) \) to get the velocity vector \( \mathbf{v}(t) \).
2. Differentiate \( \mathbf{v}(t) \) to get the acceleration vector \( \mathbf{a}(t) \).
3. Use the formulas for the tangential and normal components of acceleration:
   - \( a_T = \frac{\mathbf{v}(t) \cdot \mathbf{a}(t)}{|\mathbf{v}(t)|} \)
   - \( a_N = \frac{| \mathbf{v}(t) \times \mathbf{a}(t) |}{|\mathbf{v}(t)|} \)

Let's compute step by step.

### Step 1: Velocity vector \( \mathbf{v}(t) \)
The velocity is the derivative of \( \mathbf{r}(t) \):
\[
\mathbf{v}(t) = \frac{d}{dt} \left( t \hat{i} + 2e^t \hat{j} + e^{2t} \hat{k} \right) = \hat{i} + 2e^t \hat{j} + 2e^{2t} \hat{k}
\]

### Step 2: Acceleration vector \( \mathbf{a}(t) \)
The acceleration is the derivative of \( \mathbf{v}(t) \):
\[
\mathbf{a}(t) = \frac{d}{dt} \left( \hat{i} + 2e^t \hat{j} + 2e^{2t} \hat{k} \right) = 2e^t \hat{j} + 4e^{2t} \hat{k}
\]

### Step 3: Tangential component \( a_T \)
We first compute the dot product of \( \mathbf{v}(t) \) and \( \mathbf{a}(t) \):
\[
\mathbf{v}(t) \cdot \mathbf{a}(t) = (1)(0) + (2e^t)(2e^t) + (2e^{2t})(4e^{2t}) = 4e^{2t} + 8e^{4t}
\]

Now, compute the magnitude of the velocity vector \( \mathbf{v}(t) \):
\[
|\mathbf{v}(t)| = \sqrt{(1)^2 + (2e^t)^2 + (2e^{2t})^2} = \sqrt{1 + 4e^{2t} + 4e^{4t}}
\]

Finally, the tangential component is:
\[
a_T = \frac{4e^{2t} + 8e^{4t}}{\sqrt{1 + 4e^{2t} + 4e^{4t}}}
\]

### Step 4: Normal component \( a_N \)
We need to compute the magnitude of the cross product \( \mathbf{v}(t) \times \mathbf{a}(t) \).

Let me calculate these final components for you.The tangential (\(a_T\)) and normal (\(a_N\)) components of the acceleration vector are:

\[
a_T = \frac{(8e^{2t} + 4)e^{2t}}{\sqrt{4e^{4t} + 4e^{2t} + 1}}
\]

\[
a_N = \frac{2\sqrt{4e^{6t} + 4e^{4t} + e^{2t}}}{\sqrt{4e^{4t} + 4e^{2t} + 1}}
\]

These are the required components of the acceleration vector.

Would you like further details on any step or a more detailed explanation?

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Here are five related questions for you to consider:
1. How do the velocity and acceleration vectors behave as \(t \to \infty\)?
2. What is the geometric interpretation of the tangential and normal components?
3. Can you find the curvature of the trajectory given \( \mathbf{r}(t) \)?
4. How would the results change if the position vector were in 2D instead of 3D?
5. How can these components help in understanding the motion in physics?

**Tip:** The tangential component measures the acceleration along the path, while the normal component gives the acceleration perpendicular to the path.

Find the tangential and normal components of the acceleration vector r(t) = t i + 2e^t j + e^{2t} k

Learn how to compute the tangential (a_T) and normal (a_N) components of the acceleration vector for the parametric equation r(t) = t i + 2e^t j + e^{2t} k. This problem involves vector calculus concepts such as dot product, cross product, and acceleration components. Suitable for college calculus students.

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