Motion in Vector Form

In short: In vector form, an object's position is $\mathbf{r}(t)=x(t)\mathbf{i}+y(t)\mathbf{j}$. Differentiate each part to get the velocity and acceleration, or integrate to go back, and the speed is the magnitude of the velocity, $|\mathbf{v}|$.

Starting point: what you already know

• Vectors in component form: $\mathbf{a}=x\mathbf{i}+y\mathbf{j}$, with magnitude $|\mathbf{a}|=\sqrt{x^2+y^2}$ (Introduction to Vectors).
• Parametric equations: a path can be given as $x=f(t)$, $y=g(t)$, and you can remove $t$ to get a Cartesian equation (Parametric Equations).
• Calculus of motion from Mathematics Advanced: velocity is the derivative of displacement, $v=\frac{dx}{dt}$, and acceleration is the derivative of velocity, $a=\frac{dv}{dt}$.
• Speed is the size of the velocity, so it is never negative.

Position as a vector

Put the origin somewhere fixed. At time $t$ the object is at the point $(x(t),y(t))$, so its position vector is

$$\mathbf{r}(t)=x(t)\mathbf{i}+y(t)\mathbf{j}$$

The two parts $x(t)$ and $y(t)$ are exactly the parametric equations of the path, with time as the parameter.

• Displacement from time $t_1$ to $t_2$ is $\mathbf{r}(t_2)-\mathbf{r}(t_1)$.
• Distance between the two positions is the magnitude of that displacement.

Constant velocity

If an object starts at ${\mathbf{r}}_0$ and moves with a fixed velocity $\mathbf{v}$, then after $t$ seconds

$$\mathbf{r}(t)={\mathbf{r}}_0+t\mathbf{v}$$

Velocity and acceleration by calculus

Differentiate each part on its own:

Going the other way, integrate each part, and use the starting values to find the constants.

Relative velocity: currents and crosswinds

What an observer on the ground sees is the vector sum:

$${\mathbf{v}}_{\text{ground}}={\mathbf{v}}_{\text{boat in still water}}+{\mathbf{v}}_{\text{current}}$$

The same rule works for a plane and the wind.

Quick summary

• Position: $\mathbf{r}(t)=x(t)\mathbf{i}+y(t)\mathbf{j}$. Its parts are parametric equations of the path.
• Constant velocity: $\mathbf{r}(t)={\mathbf{r}}_0+t\mathbf{v}$.
• Differentiate each part to get velocity and acceleration. Integrate each part to go back, using starting values.
• Speed is $|\mathbf{v}|$. Distance between two positions is the size of the displacement.
• Ground velocity = own velocity + current (or wind).

Quiz