Speed and Velocity in 2 and 3 Dimensions

In these revision notes for Speed and Velocity in 2 and 3 Dimensions, we cover the following key points:

• The steps to calculate the average speed in two and three dimensions?
• How can we calculate the instantaneous speed in two and three dimensions?
• The same for the average and instantaneous velocity in two and three dimensions

Speed and Velocity in 2 and 3 Dimensions Revision Notes

The calculation of the average speed in two dimensions consists on the following steps:

1. Splitting the distance in small regular segments (exactly as we did when we were discussing about the calculation of distance in two dimensions)
2. Finding the horizontal and vertical components for each segment. The values are all positive regardless the direction as we are dealing with distance, not displacement.
3. Using the Pythagorean Theorem for calculating the distance in each segment (it represents the hypotenuse of each small right triangle formed)
4. Finding the sum of all hypotenuses obtained at (3). This gives the total distance travelled.
5. Dividing the distance found at (4) to the total time taken.

As for the instantaneous speed, we use again the same procedure as in the one-dimensional motion, i.e. after calculating the distance in a small interval around the required point (using the Pythagorean Theorem for two coordinates) we divide it by the given time interval. The result of the division gives the instantaneous speed in two dimensions.

The procedure for finding the average velocity in two dimensions is much shorter than the one used for calculating the average speed. We need only the coordinates of the initial and final position; none of the in-between coordinates are needed.

1. First we find the x and y-coordinates of the starting and ending point (xi, yi, xf, and yf)

2. Then we find the displacement for each direction by using the equations

and

3. Then we find the magnitude of the total displacement ∆r⃗ by using the equation

4. Finally, we calculate the magnitude of the average velocity using the equation

where t is the time taken for the entire motion.

For Instantaneous Velocity, the procedure is the same as in the case of instantaneous speed. The only difference is the sign of result, which unlike in the instantaneous speed, here can be positive or negative depending on the sign of the displacement.

Everything discussed for Speed and Velocity in two dimensions is also true for these quantities in three dimensions. We have only to add a new dimension (coordinate) z in the calculations.

In all situations discussed so far, we have assumed the motion is either uniform (i.e. at the same speed) or it was "modulated" in such a way that the average values of distance, displacement, speed and velocity replaced the actual values. In this way, the calculations became easier and shorter.