91Ó°ÊÓ

A moving object can be described using its position function \(\mathbf{r}(t)\), which gives the object's position at time \(t\). The velocity \(\mathbf{v}(t)\) and acceleration \(\mathbf{a}(t)\) are the first and second derivatives of the position function, respectively. The definitions are as follows: 1. Velocity: \(\mathbf{v}(t) = \frac{d}{dt}\mathbf{r}(t)\) 2. Acceleration: \(\mathbf{a}(t) = \frac{d}{dt}\mathbf{v}(t)\) The speed of the object is the magnitude of its velocity vector.

To find the object's velocity function, we need to differentiate the given position function \(\mathbf{r}\) with respect to time \(t\). The position function is a vector function, so we need to differentiate each component of the vector. If \(\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle\), then the velocity is given by: \(\mathbf{v}(t) = \frac{d}{dt}\mathbf{r}(t) = \left\langle \frac{dx(t)}{dt}, \frac{dy(t)}{dt}, \frac{dz(t)}{dt} \right\rangle\)

The speed is the magnitude of the velocity vector \(\mathbf{v}(t)\). If \(\mathbf{v}(t) = \langle u(t), v(t), w(t) \rangle\), then the speed is calculated as follows: Speed: \(|\mathbf{v}(t)| = \sqrt{u^2(t) + v^2(t) + w^2(t)}\)

Finally, to find the object's acceleration function, we need to differentiate the velocity function we obtained earlier. Just like we did before, we need to differentiate each component of the velocity vector. If \(\mathbf{v}(t) = \langle u(t), v(t), w(t) \rangle\), then the acceleration is given by: \(\mathbf{a}(t) = \frac{d}{dt}\mathbf{v}(t) = \left\langle \frac{du(t)}{dt}, \frac{dv(t)}{dt}, \frac{dw(t)}{dt} \right\rangle\) In summary, we found the velocity, speed, and acceleration of the moving object by using the given position function \(\mathbf{r}(t)\). The velocity is the first derivative of the position function, the acceleration is the second derivative (or the first derivative of the velocity function), and the speed is the magnitude of the velocity vector.