3.1 Kinematics in two dimensions: an introduction

Learning objectives

By the end of this section, you will be able to:

• Observe that motion in two dimensions consists of horizontal and vertical components.
• Understand the independence of horizontal and vertical vectors in two-dimensional motion.

The information presented in this section supports the following AP® learning objectives and science practices:

• 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, and graphical representations. (S.P. 1.5, 2.1, 2.2)
• 3.A.1.2 The student is able to design an experimental investigation of the motion of an object. (S.P. 4.2)
• 3.A.1.3 The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations. (S.P. 5.1)

Two-dimensional motion: walking in a city

Suppose you want to walk from one point to another in a city with uniform square blocks, as pictured in [link] .

The straight-line path that a helicopter might fly is blocked to you as a pedestrian, and so you are forced to take a two-dimensional path, such as the one shown. You walk 14 blocks in all, 9 east followed by 5 north. What is the straight-line distance?

An old adage states that the shortest distance between two points is a straight line. The two legs of the trip and the straight-line path form a right triangle, and so the Pythagorean theorem, $a^2\text{ + }{b}^2\text{ = }{c}^2$ , can be used to find the straight-line distance.

The hypotenuse of the triangle is the straight-line path, and so in this case its length in units of city blocks is $\sqrt{(\text{9 blocks}{)}^2\text{+ }(\text{5 blocks}{)}^2}\text{= 10}\text{.}\text{3 blocks}$ , considerably shorter than the 14 blocks you walked. (Note that we are using three significant figures in the answer. Although it appears that “9” and “5” have only one significant digit, they are discrete numbers. In this case “9 blocks” is the same as “9.0 or 9.00 blocks.” We have decided to use three significant figures in the answer in order to show the result more precisely.)

The fact that the straight-line distance (10.3 blocks) in [link] is less than the total distance walked (14 blocks) is one example of a general characteristic of vectors. (Recall that vectors are quantities that have both magnitude and direction.)