Kinematics in One Dimension

Position Vector

Displacement Vector, DistanceVelocity Vector

Acceleration Vector

Kinematics

Geometrical and algebraic description of motion

No regard to the causes of motion (forces)

Makes use of the mathematical concept of coordinate system attached to a point in space or to objects called ‘frames of reference’.

Coordinate system is used to describe position relative to the origin.

Location of origin is arbitrarily selected.

One Dimensional Coordinate System When dealing with a single dimension the coordinate

system is a straight line. Points on the line are located relative to an arbitrary

point called the origin. Distance on one side of the origin is positive; negative on

the opposite side. A point in 1D space corresponds to a point on the line; a

positive or negative number with units of length. Positive and negative axis directions are arbitrarily selected; usually the positive direction is chosen to the right. Thus a point can be described by a single signed coordinate value with unit.

Sample point is 1.5 m on the positive side of the origin.

O +1 +2-1-2x (m)

+1.5 m

Position as a Vector

A point can also be located by a position vector.

Position vector has tail at the origin and tip at the point.

The length of the vector is the distance (coordinate value) of the point from the origin.

The vector direction corresponds to the sign of the coordinate value.

O +1 +2-1-2x (m)

+1.5 m

Displacement Vector - describes the change in position of a moving object Vector pointing from an object’s initial

position to its final position.

0

0

xxx

x

x

Vectornt Displaceme

ectorposition v final

ectorposition v initial

In one dimension position and displacement, although vectors, can be unambiguously described by scalars.

axis.- negativeor positive

alongeither asnt displaceme

ofdirection indicates ofSign

ntdisplaceme

coordinateposition final

coordinateposition initial

0

0

x

x

xxx

x

x

Displacement has unit of length such as m.

Positive and Negative Displacement

Distance

Magnitude of displacement if motion is in one direction only (no reversal of direction).

Distance is always positive because it is has magnitude only with no direction.

If motion consists of a sequence of positive and negative displacements, the distance is the sum of distances for each of the segments of the displacement.

4.0 m

4.0 m

2.0 m

x = + 4.0 m

d = 4.0 mx = + 2.0 m

d = 4.0m + 2.0 m = 6.0 m

Distance versus Displacement

What if you travel around the running track?

What is your displacement? What is your distance?

Average Speed - a scalar quantity

A stage of the Tour de France from Melun to Paris has a distance of 140 km.

Australian, Robbie McEwen, won the stage by cycling the distance in 3h, 30 min and 47 s. What was his average speed?

km/h 9.39851.39h 513.3

km 140

h 513.3

min 60

h 1

s 60

min 1s 47

min 60

h 1min 30h 3

t

dv

t

t

speed

t

dvspeed

Speed has units of length per unit time such as m/s.

Average Velocity - a vector quantity

t

xv

A man taking a leisurely walk takes 15 minutes to walk 100 m in an eastward direction. He then walks 50 m back (west) in 5 minutes. What is his average speed and velocity in m/s?

m/s 12.0

min 1

s 60min 5 min 15

m 50 m 100

Speed Average

speed

speed

v

t

dv

(east) m/s 042.0s 1200

m 50

s 1200min 20t

(east) m 50m 50 - m 100

positive. asdirection eastward Take

t

xv

x

EW100 m

50 m

Velocity has units of length per unit time such as m/s.

Instantaneous Velocity - average velocity when the elapsed time approaches zero

t

xv

0

limt

Instantaneous Velocity or simply velocity

Instantaneous Speed or simply speed - the magnitude of the velocity.

If v is constant, the x-t graph is that of a straight line. The average and instantaneous velocities are the same and is equal to the slope of the line. If v is not constant the instantaneous velocity at a time t is the slope of the curved x-t graph. Instantaneous and average velocities are not necessarily the same (although they can be).

Small intervals give you instants

Meaning of the sign of velocity

Positive velocity means direction of v is towards positive direction of x axis, i.e., displacement is positive. Slope of x-t graph will be positive.

Negative velocity means direction of v is towards negative x axis, i.e., displacement is negative. Slope of x-t graph will be negative.

The sign of v does not indicate whether speed is increasing or decreasing. It, however, indicates what is happening to the displacement.

Meaning of the sign of velocity

•You can consider the world to have a positive direction and a negative direction.

•Are you going towards POSIWORLD?

•Or NEGILAND?

Acceleration - a vector quantity

Average Acceleration

t

va

avgUnit: length/time2 such as m/s2

Instantaneous Acceleration - average acceleration when the elapsed time approaches zero.

t

va

0

limt

Acceleration and Velocity

Whether an object is speeding up or slowing down does not depend on the sign of a but on its direction relative to the direction of v.

a v Result (assume + )

v is increasing in magnitudeand becoming more positive;speeding up

v is decreasing in magnitudeand become less negative;slowing down

v is decreasing in magnitudeand becoming less positive;slowing down

v is increasing in magnitudeand becoming more negative;speeding up

a and v in the same direction

0 assumed is timeinitial if 0

0

0

0

atvv

tavv

vvtav

vvvvvt

va

if

a and v in opposite directions

0 assumed is timeinitial if

m/s 131528

s 3m/s 5-m/s 28

0

20

0

0

atvv

v

tavv

vvtav

vvvvvt

va

if

t = 3 s

a and v in opposite directions

t = 3 s

When acceleration and velocity ‘compete’ then the object is slowing down

Kinematic Equations for Constant Acceleration

We already have one kinematic equation (previous slide)atvv 0

The graph of v vs. t if a is constant is a straight line.

t

v

v0 slope = a

mxby

vavg

0

0

0

2

tt

xx

t

xv

vvvvavg

Usually, we assume initial time t0 = 0 and initial position x0 = 0.

20

000

2

1

22

attvx

tvatv

tvv

tvx

Kinematic Equations . . .

axvv

axvv

a

vvx

a

vvvvt

vvx

a

vvtatvv

2

2

2

22

or

20

2

20

2

20

2

000

00

Kinematics Equations

Summary of Kinematic Equations in 1 D constant a

Variables Equation x a v v0 t

atvv 0

tvvx 021

221

0 attvx

axvv 220

2

Remember, kinematic variables are vector quantities. Their signs are important. Also x is a displacement from the initial position which was assumed to be zero. Initial time was also assumed zero.

Freely Falling BodiesExample of motion with constant acceleration.

Motion is vertical with acceleration of gravity,

g = 9.81 m/s2 always downward.

Coordinate axis will be vertical (you can call it x or y). Choose positive direction up or down but be sure the sign of a is correct.

If you choose upward is positive, then a = -g.

If you choose downward as positive, then a = +g.

Be sure to get study packet and we will work some problems out in class.