Conceptual Physics

4.1 - Velocity in two dimensions

Velocity is a vector quantity, meaning it contains two pieces of information: how fast

something is traveling and in which direction. Both are crucial for understanding motion

in multiple dimensions.

Consider the car on the track to the right. It starts out traveling parallel to the x axis at a

constant speed. It then reaches a curve and continues to travel at a constant speed

through the curve. Although its speed stays the same, its direction changes. Since

velocity is defined by speed and direction, the change in direction means the car’s

velocity changes.

Using vectors to describe the car’s velocity helps to illustrate its change in velocity. The

velocity vector points in the direction of the car’s motion at any moment in time. Initially,

the car moves horizontally, and its velocity vector points to the right, parallel to the x

axis. As the car goes around the curve, the velocity vector starts to point upward as well

as to the right. You see this shown in the illustration for Concept 1.

When the car exits the curve, its velocity vector will be straight up, parallel to the y axis.

Because the car is moving at a constant speed, the length of the vector stays the same:

The speed, or magnitude of the vector, remains constant. However, the direction of the

vector changes as the car moves around the curve.

Like any vector, the velocity vector can be written as the sum of its components, the

velocities along the x and y axes. This is also shown in the Concept 1 illustration. The

gauges display the x and y velocities. If you click on Concept 1 to see the animated

version of the illustration, you will see the gauges constantly changing as the car rounds

the bend. At the moment shown in the illustration, the car is moving at 17 m/s in the

horizontal direction and 10 m/s in the vertical direction. The components of the vector

shown also reflect these values. The horizontal component is longer than the vertical

one.

Equations 1 and 2 show equations useful for analyzing the car’s velocity. Equation 1

shows how to break the car’s overall velocity into its components. (These equations

employ the same technique used to break any vector into its components.) The

illustration shows the car’s velocity vector. The angle ș is the angle the velocity vector

makes with the positive x axis. The product of the cosine of that angle and the

magnitude of the car’s velocity (its speed) equals the car’s horizontal velocity

component. The sine of the angle times the speed equals the vertical velocity

component.

The first equation in Equation 2 shows how to calculate the car’s average velocity when

its displacement and the elapsed time are known. The displacement ǻr divided by the

elapsed time ǻt equals the average velocity.

The equations for determining the average velocity components when the components

of the displacement are known are also shown in Equation 2. Dividing the displacement

along the x axis by the elapsed time yields the horizontal component of the car’s

average velocity. The displacement along the y axis divided by the elapsed time equals

the vertical component of the average velocity. A demonstration of these calculations is

shown in Example 1.

The distinction between average and instantaneous velocity parallels the discussion of these two topics in the study of motion in one

dimension. To determine the instantaneous velocity, ǻr is measured during a very short increment of time and divided by that increment.

As with linear motion, the velocity vector points in the direction of motion. On the curved part of the track, the instantaneous velocity vector is

tangent to the curve, since that is the direction of the car’s motion at any instant in time. The average velocity vector points in the same

direction as the displacement vector used to determine its value.

Velocity in two dimensions

Velocity has x and y components

Analyze x and y components separately

Two component vectors sum to equal

total velocity

Components of velocity

vx = v cos ș

vy = v sin ș

v = speed

ș = angle with positive x axis

vx = x component of velocity

vy = y component of velocity

(^66) Copyright 2000-2007 Kinetic Books Co. Chapter 04