Section 17.2Position, Velocity, and Acceleration

Position• The “position vector” of an object is defined by

the vector valued function (where t is time)

• For example

• This gives us a path that is a cylindrical helix with a radius of 10 and 2 windings up around the z-axis

• Our position vector is measured in whatever units each component is measured in– For our example we will use feet

btathtgtftr where)(),(),()(

40where,sin10,cos10)( tttttr

Instantaneous Velocity• If is a position vector, than the velocity vector

is defined to be

• So for our previous example we have

• Note: Instantaneous speed is given by

• The velocity vector is tangent to the object’s path

• What is the speed for the above helix?

)(tr

)(),(),()()( thtgtftrtv

ft/secinmeasured1,cos10,sin10)(

40where,sin10,cos10)(

tttv

tttttr

222 )]('[)]([)]([)()()( thtgtftvtvtv

Acceleration

• The acceleration vector is given by

• The magnitude of the acceleration vector gives us the magnitude of the net force acting on an object

)(),(),()()()( thtgtftrtvta

2

( ) 10cos ,10sin , where 0 4

( ) 10sin ,10cos ,1 measured in ft/sec

10cos , 10sin ,0 measured in ft/sec

r t t t t t

v t t t

a(t) t t

Uniform Circular Motion• When an object travels in a circular path with a

constant speed we call it uniform circular motion

• Its motion can be described by

• Motion is in a circle of radius R with period 2π/ω

• Velocity vector is tangent to circle and speed is constant with

• Acceleration vector points toward center of circle with

• Let’s take a look at our above function with Maple

jtRitRtr

)sin()cos()(

Rv

Rva /2

Uniform Straight Line Motion

• What do we need in order to ensure we have straight line motion?

• For a particle whose motion is described by

• Motion is along a straight line through the point with position vector parallel to

• Both the velocity and acceleration vectors are parallel to the line

• Let’s take a look at #14 with Maple

00 )()( vtfrtr

0r

0v

Length of a Curve• Now we know that the speed of an object is

• Then just as in one dimension we can find the distance traveled (i.e. length of path or curve) by integrating its speed

222

Speed

dt

dz

dt

dy

dt

dxv

dttvb

a )(CurveofLength