## Motion in Two and Three Dimensions

### Position in Two and Three Dimensions

Extending position to two and three dimensions mostly requires handling vector notation. The position is given by: $$\vec{r}(t) = x(t)\, \hat{i} + y(t)\, \hat{j} + z(t)\, \hat{k} $$ Here \(x(t)\), \(y(t)\), and \(z(t)\) are the time-dependent coordinates of the position vector. Each acts exactly like the one-dimensional position function. The value gives the magnitude of the component and the sign indicates the direction along each axis.

### Velocity in Two and Three Dimensions

The velocity function is the time-derivative of the position function.

Really this is it. The rest is understanding how vectors work. The motion in the separate dimensions should be kept separate.### A Great Question About 2D Motion

This question is so good, it gets its own topic area.

I'm throwing a ball to two of my kids, Alice and Bob (not their real names), at the same time. The trajectories are shown in the figure. Which is true?

- The ball to Alice arrives first.
- The ball to Bob arrives first.
- Both balls arrive at the same time.
- Need more information to know arrival times.

Make sure you decide on an answer. The answer and discussion are in the video.

### Acceleration in Two and Three Dimensions

The acceleration function is the time-derivative of the velocity function.

Notice a pattern? The acceleration is telling you how the velocity is changing. The velocity can change magnitude or direction or both, these changes are caused by accelerations.### Exercises

*Do Now!* Do these exercises immediately.

*Not Now!* Do these after you start to forget the topic, say in a week.

*More!* More exercises if you want. Maybe review before a test.

### Required Problems

### Solution

1. The acceleration of a particle as a function of time is given by \(\vec{a} = 6t^2\, \hat{i} + 4\cos (2\pi t)\, \hat{j}\). The position and velocity are zero at \(t=0\). What are the velocity and position functions? What is the magnitude of the velocity at \(t=2\)? What is the magnitude of the position at \(t=1/2\)?