We are tasked with finding the tangential and normal components of the acceleration for a particle whose position is given by:

$\vec{r}(t) = \left( t^2, -2t, \ln t \right)$

To find the tangential component $a_T$ and the normal component $a_N$ of the acceleration at $t = 1$, we will follow these steps:

1. Velocity $\vec{v}(t)$

The velocity is the derivative of the position vector:

$\vec{v}(t) = \frac{d\vec{r}(t)}{dt} = \left( \frac{d}{dt}(t^2), \frac{d}{dt}(-2t), \frac{d}{dt}(\ln t) \right)$ $\vec{v}(t) = (2t, -2, \frac{1}{t})$

At $t = 1$:

$\vec{v}(1) = (2(1), -2, \frac{1}{1}) = (2, -2, 1)$

2. Acceleration $\vec{a}(t)$

The acceleration is the derivative of the velocity vector:

$\vec{a}(t) = \frac{d\vec{v}(t)}{dt} = \left( \frac{d}{dt}(2t), \frac{d}{dt}(-2), \frac{d}{dt}(\frac{1}{t}) \right)$ $\vec{a}(t) = (2, 0, -\frac{1}{t^2})$

At $t = 1$:

$\vec{a}(1) = (2, 0, -1)$

3. Tangential Component $a_T$

The tangential component of the acceleration is given by:

$a_T = \frac{\vec{v}(t) \cdot \vec{a}(t)}{|\vec{v}(t)|}$

First, calculate the dot product $\vec{v}(1) \cdot \vec{a}(1)$:

$\vec{v}(1) \cdot \vec{a}(1) = (2)(2) + (-2)(0) + (1)(-1) = 4 - 1 = 3$

Now, calculate the magnitude of $\vec{v}(1)$:

$|\vec{v}(1)| = \sqrt{(2)^2 + (-2)^2 + (1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3$

Thus, the tangential component is:

$a_T = \frac{3}{3} = 1$

4. Normal Component $a_N$

The normal component of the acceleration is given by:

$a_N = \sqrt{|\vec{a}(t)|^2 - a_T^2}$

First, calculate the magnitude of $\vec{a}(1)$:

$|\vec{a}(1)| = \sqrt{(2)^2 + (0)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}$

Now, calculate $a_N$:

$a_N = \sqrt{|\vec{a}(1)|^2 - a_T^2} = \sqrt{(\sqrt{5})^2 - (1)^2} = \sqrt{5 - 1} = \sqrt{4} = 2$

5. Conclusion

The tangential component $a_T = 1$ and the normal component $a_N = 2$.

Thus, the correct answer is (c) $a_T = 1, a_N = 2$.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How do you find the unit tangent vector in vector calculus?
2. What is the physical meaning of tangential and normal components of acceleration?
3. How do you compute curvature from the position vector of a particle?
4. How does the magnitude of velocity relate to tangential acceleration?
5. Can the normal component of acceleration ever be zero?

Tip: When solving problems with vectors, always remember to differentiate each component individually and apply the chain rule when necessary.