Can you help me with this one **Problem Statement:**Find the tangential component \( a_T \) and the normal component \( a_N \) of acceleration as a function of \( t \) if \( \mathbf{r}(t) = \langle 2t^2, 7t^3 \rangle \).\( a_T(t) = \, ? \)\( a_N(t) = \, ? \)**Explanation:**This problem involves finding components of acceleration for the vector function \( \mathbf{r}(t) \). The vector \( \mathbf{r}(t) \) describes a position in terms of the parameter \( t \), with components along \( x \) and \( y \) axes given by \( 2t^2 \) and \( 7t^3 \), respectively.To solve this:1. **Calculate the velocity vector \( \mathbf{v}(t) \):** - **Derivative of position vector:** \[ \mathbf{v}(t) = \frac{d}{dt}\langle 2t^2, 7t^3 \rangle = \langle 4t, 21t^2 \rangle \]2. **Calculate the acceleration vector \( \mathbf{a}(t) \):** - **Derivative of velocity vector:** \[ \mathbf{a}(t) = \frac{d}{dt}\langle 4t, 21t^2 \rangle = \langle 4, 42t \rangle \]3. **Find the magnitudes:** - **Speed \( v(t) \):** \[ v(t) = \|\mathbf{v}(t)\| = \sqrt{(4t)^2 + (21t^2)^2} = \sqrt{16t^2 + 441t^4} \]4. **Calculate the tangential component \( a_T(t) \):** - **Using the formula \( a_T = \frac{d}{dt}v(t) \):**5. **Calculate the normal component \( a_N(t) \):** - **Using the formula \( a_N = \sqrt{\|\mathbf{a}(t)\|^2 - a_T^2} \):**This involves further differentiating and simplifying to arrive

Name: Can you help me with this one **Problem Statement:**Find the tangentia
Uploaded: 2024-03-20T07:07:26.000Z
Channel: Bartleby
Description: Can you help me with this one **Problem Statement:**Find the tangential component \( a_T \) and the normal component \( a_N \) of acceleration as a function of \( t \) if \( \mathbf{r}(t) = \langle 2t^2, 7t^3 \rangle \).\( a_T(t) = \, ? \)\( a_N(t) =