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Chapter 12: Problem 11

Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. $$\mathbf{r}(t)=\langle 8 \sin t, 8 \cos t\rangle, \text { for } 0 \leq t \leq 2 \pi$$

Short Answer

Expert verified
Answer: The velocity of the object is \(\mathbf{v}(t) = \langle 8 \cos t, -8 \sin t \rangle\), the speed is 8, and the acceleration is \(\mathbf{a}(t) = \langle -8 \sin t, -8 \cos t \rangle\).

Step by step solution

01

Calculate the velocity vector

To find the velocity vector, we take the derivative of the position vector with respect to time: $$\mathbf{v}(t) = \frac{d \mathbf{r}(t)}{dt} = \frac{d \langle 8 \sin t, 8 \cos t \rangle }{dt}$$ Differentiate each component separately: $$\mathbf{v}(t) = \left\langle \frac{d(8 \sin t)}{dt}, \frac{d(8 \cos t)}{dt} \right\rangle$$ Now apply the chain rule: $$\mathbf{v}(t) = \langle 8 \cos t, -8 \sin t \rangle$$
02

Calculate the speed

The speed of an object is the magnitude of the velocity vector. Therefore, we need to find the magnitude of the velocity vector \(\mathbf{v}(t) = \langle 8 \cos t, -8 \sin t \rangle\). We do this using the Pythagorean theorem: $$\|\mathbf{v}(t)\| = \sqrt{(8 \cos t)^2 + (-8 \sin t)^2}$$ Simplify and factor out a common term: $$\|\mathbf{v}(t)\| = \sqrt{64(\cos^2 t + \sin^2 t)}$$ Since \(\cos^2 t + \sin^2 t = 1\), then the speed is: $$\|\mathbf{v}(t)\| = 8$$
03

Calculate the acceleration vector

To find the acceleration vector, we take the derivative of the velocity vector with respect to time: $$\mathbf{a}(t) = \frac{d \mathbf{v}(t)}{dt} = \frac{d \langle 8 \cos t, -8 \sin t \rangle }{dt}$$ Differentiate each component separately: $$\mathbf{a}(t) = \left\langle \frac{d(8 \cos t)}{dt}, \frac{d(-8 \sin t)}{dt} \right\rangle$$ Now apply the chain rule: $$\mathbf{a}(t) = \langle -8 \sin t, -8 \cos t \rangle$$ Now we have found the velocity, speed, and acceleration of the object: - Velocity: \(\mathbf{v}(t) = \langle 8 \cos t, -8 \sin t \rangle\) - Speed: \(\|\mathbf{v}(t)\| = 8\) - Acceleration: \(\mathbf{a}(t) = \langle -8 \sin t, -8 \cos t \rangle\)

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