Transcript Motion Vectors

Motion Vectors
Displacement Vector

The position vector
 is often
designated by r .
y


A change in position is a
displacement.
The displacement vector is a
change in the position
vector.

r1

r2
  
r  r2  r1
x
Time is a Scalar
y
t1

Position and displacement
are vectors.

Time has a value but no
direction – it’s a scalar.

Vector quantities can be
multiplied or divided by time
using scalar multiplication.
t  t2  t1
t2
x
Velocity Vector


Displacement divided by
time gives the average
velocity.
The displacement vector
divided by time gives the
average velocity vector.

A person walks from 2.0 km
north of the gym to 3.0 km west
of the gym in 1.5 h. What is the
magnitude of average velocity?
• The magnitude of
displacement is
d  d x2  d y2  3.6 km


r
vav 
t
• The average velocity is the
magnitude of displacement
divided by the time.
vav  d / t  2.4 km/h
Tangent



The average velocity
becomes the instantaneous
velocity for short time
intervals.
The same is true for vectors.
The instantaneous velocity
vector direction is tangent to
the curve.

 dr
v
dt
y

r1

v

r2
x
Derivative of a Vector

The derivative of a vector can be computed as a
derivative of the separate components.

 dr dx ˆ dy ˆ
v
 i
j
dt dt
dt
dx
dt
dy
vy 
dt
vx 
Velocity Graph

Position graph with velocity
vectors

Velocity graph
y
vy

r1

v1

v
 2
r2
x

v2

v1
vx
Acceleration Vector

Average acceleration is the
ratio of the change in
velocity to the time.

Instantaneous acceleration
is the derivative of the
velocity with respect to time.

Both these definitions extend
to vectors.
  

v v2  v1
aav 

t
t


 dv d 2 r
a
 2
dt dt
Vector Equations

Like velocity, acceleration equations can be written
by components.

 dv dvx ˆ dv y ˆ
a

i
j
dt
dt
dt
dvx d 2 x
ax 
 2
dt
dt
dv y d 2 y
ay 
 2
dt
dt
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