calculus-based physics for scientists and engineers

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?

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

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

person throwing two balls

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.


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


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 \(t=1/2\)?

Optional Problems