The position vector r describes the path of an object moving in space. Position Vector Time r(t) = 3ti + tj + k t = 4 (a) Find the velocity vector v(t), speed s(t), and acceleration vector a(t) of the object. v(t) = s(t) a(t) = (b) Evaluate the velocity vector and acceleration vector of the object at the given value of v(4) = a(4) =
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
12.3q2
![**Topic: Understanding Position, Velocity, and Acceleration Vectors in Space**
---
The position vector \( \mathbf{r} \) describes the path of an object moving in space.
### Position Vector
\[ \mathbf{r}(t) = 3t \mathbf{i} + t \mathbf{j} + \frac{1}{8}t^2 \mathbf{k} \]
### Time
\[ t = 4 \]
#### Questions:
(a) **Find the velocity vector \( \mathbf{v}(t) \), speed \( s(t) \), and acceleration vector \( \mathbf{a}(t) \) of the object.**
\[ \mathbf{v}(t) = \boxed{\hspace{100px}} \]
\[ s(t) = \boxed{\hspace{100px}} \]
\[ \mathbf{a}(t) = \boxed{\hspace{100px}} \]
(b) **Evaluate the velocity vector and acceleration vector of the object at the given value of \( t \).**
\[ \mathbf{v}(4) = \boxed{\hspace{100px}} \]
\[ \mathbf{a}(4) = \boxed{\hspace{100px}} \]
---
To proceed with the calculations:
- The velocity vector \( \mathbf{v}(t) \) is obtained by differentiating the position vector \( \mathbf{r}(t) \) with respect to time \( t \).
- The speed \( s(t) \) is the magnitude of the velocity vector \( \mathbf{v}(t) \).
- The acceleration vector \( \mathbf{a}(t) \) is obtained by differentiating the velocity vector \( \mathbf{v}(t) \) with respect to time \( t \).
### Steps:
1. **Calculate \( \mathbf{v}(t) \)**:
\[ \mathbf{v}(t) = \frac{d}{dt}[3t \mathbf{i} + t \mathbf{j} + \frac{1}{8}t^2 \mathbf{k}] \]
2. **Calculate \( s(t) \)**:
\[ s(t) = |\mathbf{v}(t)| \]
3. **Calculate \( \mathbf{a}(t) \)**:
\[ \mathbf{a}(t) = \frac{d}{dt}\mathbf{v}(t](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fafc8379c-b6a2-4db8-9fce-fab4471db5da%2Facb0bb1b-86f4-4f0b-b872-b67bb43a7b6a%2Fhd9r0hs_processed.png&w=3840&q=75)
Step by step
Solved in 2 steps with 2 images
