8.4: Motion in Space

1. The position of a particle at time $t$ is given by the function $\mathbf{r}(t) = (3t^2 - 2, 2t - \sin t)$. Find the velocity, acceleration, and speed of this particle in terms of $t$.
    
2. The position of a particle at time $t$ is given by the function $\mathbf{r}(t) = 2\sin t\,\mathbf{i} + 2\cos t\,\mathbf{j} + t^2\,\mathbf{k}$. Find the velocity, acceleration, and speed of the particle in terms of $t$.
    
3. The position of a particle at time $t$ is given by the function $\mathbf{r}(t) = (e^{-t}, t^2, \tan t)$. Find the velocity, acceleration, and speed of the particle in terms of $t$.
    
4. A projectile is launched into the air from ground level with an initial speed of 50 m/sec at an angle of 60° with the horizontal. Construct a vector-valued function that models the position of the projectile $t$ seconds after it is launched.
    
5. A ball is thrown horizontally off the edge of a 540 m tall skyscraper with an initial speed of 40 m/sec. Construct a vector-valued function that models the position of the projectile $t$ seconds after it is launched.
    
6. A projectile is launched 1.5 m above the surface of Mars with an initial speed of 100 m/sec at an angle of 45° with the horizontal. Given that the acceleration due to gravity on Mars is $g = -3.71 \text{m}/\text{sec}^2$, construct a vector-valued function that models the position of the projectile $t$ seconds after it is launched.
    
7. The position of a particle at time $t$ is given by the function $\mathbf{r}(t) = \cos t\,\mathbf{i} + \sin t\,\mathbf{j} + t\,\mathbf{k}$. Find the tangential and normal components of acceleration when $t = 0$.
    
8. The position of a particle at time $t$ is given by the function $\mathbf{r}(t) = \left(2t, t^2, \dfrac{t^3}{3}\right)$. Find the tangential and normal components of acceleration when $t = 1$.
    
9. The position of a particle at time $t$ is given by the function $\mathbf{r}(t) = 2t^3\,\mathbf{i} + 3t^2\,\mathbf{j} + 6t\,\mathbf{k}$. Find the tangential and normal components of acceleration at $t = -1$.