Transcript Displacement

Displacement
Defining Position

Position has three properties:
• Origin, magnitude, direction
1 dimension
12 feet above sea
level.




Origin: sea level
Magnitude: 12
feet
Direction: up
2 dimensions
65 miles west of
Chicago.




Origin:
downtown
Chicago
Magnitude: 65
miles
Direction: west
3 dimensions
Range 200 m,
bearing 270,
at 30 altitude.




Origin: observer
Magnitude: 200
meters
Direction: 270
by the compass
and 30 up.
Position Graph

Position can be displayed on
a graph.
• The origin for position is the
origin on the graph.
• Axes are position
coordinates.
• The position is a vector.
trajectory
y
position
vector

r
x

A set of position points
connected on a graph is a
trajectory.
2-dimensions
(x, y)
Scalar Multiplication

A vector can be multiplied by
a scalar.
• Change feet to meters.
• Walk twice as far in the
same direction.

Scalar multiplication
multiplies each component
by the same factor.

The result is a new vector,
always parallel to the original
vector.


T  sA  ( sAx , sAy )
Reference Point
trajectory

displacement
Displacement is different
from position
• Position is measured
relative to an origin common
to all points.
• Displacement is measured
relative to the object’s initial
position.
• The path (trajectory) doesn’t
matter for displacement.
position
origin
Displacement Vector

The position vector
 is often
designated by r .
y


A change in a quantity is
designated by Δ (delta).
Always take the final value
and subtract the initial value.

r1

r2
  
r  r2  r1
x
Two Displacements

A hiker starts at a point 2.0 km east of camp, then
walks to a point 3.0 km northeast of camp. What is
the displacement of the hiker?

Each individual displacement is a vector that can be
represented by an arrow.
3.0 km
2.0 km
Vector Subtraction


To subtract two vectors, place both at the same
origin.
Start at the tip of the first and go to the tip of the
second.

B

D

A
  
D  B A
Component Subtraction


Multiplying a vector by 1 will create an antiparallel
vector of the same magnitude.
Vector subtraction is equivalent to scalar
multiplication and addition.

   
D  B  A  B  (1) A

D

A

B
Dx  Bx  Ax  Bx  (1) Ax
D y  B y  Ay  B y  (1) Ay

 
D  B  (1) A
Displacement Components


B

A
  
D  B A

D
Find the components of each
vector, and subtract.
•
•
•
•
Ax = 2.0 km
Ay = 0.0 km
Bx = (3.0 km)cos45 = 2.1 km
By = (3.0 km)sin45 = 2.1 km
• Dx = Bx – Ax = 0.1 km
• Dy = By – Ay = 2.1 km
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