13.2 - Integrals of Vector Functions; Projectile Motion

Definite Integral of a Vector-Valued Function

$\int_a^b \mathbf{r}(t)\,dt = \begin{pmatrix} \int_a^b f(t)\,dt \\ \int_a^b g(t)\,dt \\ \int_a^b h(t)\,dt \end{pmatrix}$ where $\mathbf{r}(t) = \begin{pmatrix} f(t) \\ g(t) \\ h(t) \end{pmatrix}$

Fundamental Theorem of Calculus for Vector Functions

$\int_a^b \mathbf{r}(t)\,dt = [\mathbf{R}(t)]_a^b = \mathbf{R}(b) - \mathbf{R}(a)$ where $\mathbf{R}'(t) = \mathbf{r}(t)$

General Antiderivative of a Vector Function

$\int \mathbf{r}(t)\,dt = \mathbf{R}(t) + \mathbf{C}$ where $\mathbf{R}'(t) = \mathbf{r}(t)$ and $\mathbf{C} = \begin{pmatrix} C_1 \\ C_2 \\ C_3 \end{pmatrix}$ is a constant vector

Velocity Vector from Acceleration

$\mathbf{v}(t) = \int \mathbf{a}(t)\,dt + \mathbf{C}_1$ where $\mathbf{C}_1$ is determined using the initial condition $\mathbf{v}(0) = \mathbf{v}_0$

Position Vector from Velocity

$\mathbf{r}(t) = \int \mathbf{v}(t)\,dt + \mathbf{C}_2$ where $\mathbf{C}_2$ is determined using the initial condition $\mathbf{r}(0) = \mathbf{r}_0$

Projectile Motion Acceleration Vector

$\mathbf{a}(t) = \begin{pmatrix} 0 \\ -g \end{pmatrix}$ where $g = 9.8\,m/s^2$

Projectile Motion Initial Velocity Vector

$\mathbf{v}(0) = \begin{pmatrix} v_0 \cos(\alpha) \\ v_0 \sin(\alpha) \end{pmatrix}$ where $v_0$ is the launch speed and $\alpha$ is the launch angle

Projectile Motion Velocity Vector

$$\mathbf{v}(t) = \begin{pmatrix} v_0 \cos(\alpha) \\ v_0 \sin(\alpha) - g t \end{pmatrix}$$

Projectile Motion Position Vector

$$\mathbf{r}(t) = \begin{pmatrix} x_0 + (v_0 \cos(\alpha)) t \\ y_0 + (v_0 \sin(\alpha)) t - \frac{1}{2} g t^2 \end{pmatrix}$$

Time to Maximum Height in Projectile Motion

$t = \frac{v_0 \sin(\alpha)}{g}$ which occurs when the vertical velocity component satisfies $v_y(t) = 0$

Horizontal Range in Projectile Motion

$x(t) = (v_0 \cos(\alpha)) t$ evaluated at the time of flight $t$ where the vertical position component satisfies $y(t) = 0$

$$\mathbf{r}(t)$$

Position vector at time $t$

$$\mathbf{v}(t)$$

Velocity vector at time $t$

$$\mathbf{a}(t)$$

Acceleration vector at time $t$

$$\mathbf{R}(t)$$

Antiderivative vector of $\mathbf{r}(t)$

$$\mathbf{C}$$

Constant vector of integration

$$g$$

Acceleration due to gravity ($9.8\,m/s^2$)

$$v_0$$

Initial launch speed (magnitude)

$$\alpha$$

Initial launch angle (elevation)

$$\mathbf{v}_0$$

Initial velocity vector at time $t = 0$

$$\mathbf{r}_0$$

Initial position vector at time $t = 0$

$$t$$

Elapsed time parameter

$$x_0$$

Initial horizontal position at time $t = 0$

$$y_0$$

Initial vertical position at time $t = 0$

$$v_y(t)$$

Vertical velocity component at time $t$