Chapter 4 ⅗
Relative Motion & 2D Kinematics
DEHS 2012-13
Physics 1
Frame of Reference
• A frame of reference is a coordinate system in
which to measure the position, velocity,
acceleration, etc of objects in it
• Two objects are considered to be in the same
frame of reference if they are moving with the
same velocity (same speed and direction) or
rotating about the same axis
• This course will only cover frames of reference
that move with constant velocity (that do not
accelerate or rotate)
Notation for Relative Velocity
• Take note of the notation system
The velocity of
object/frame a
r
v ab
relative to object/frame b
• Think of it as object/frame b as being at rest
• Ex: a car traveling 25 mph north moves relative
r
to the ground v cg  25 mph North
Generalized Relative Motion Eqns
• Reversing the subscripts reverses the direction
r
r
of the velocity: v ab   v ba
r
r
r
• The general equation is v 13  v 12  v 23
where 1 & 3 are the objects and 2 can be

anything

Example 4.5-1
A person climbs up a ladder on a moving train with a velocity of 3 m/s
relative to the train. If the train moves relative to the ground with a
velocity of 5 m/s, what is the velocity of the person relative to an
observer watching from the ground?
Example 4.5-2
You are riding in a boat whose speed relative to the water is 6.1 m/s. The boat
points at an angle of 25° upstream on a river flowing 1.4 m/s. (a) What is
your velocity relative to the ground? (b) Suppose the speed of the boat
relative to the water remains the same, but the direction in which it points
is changed. What angle is required for the boat to go straight across the
stream? (c) What is the boat’s speed relative to the ground be?
Example 4.5-3
A couple is shopping on the 2nd floor of the mall. The wife wants to go
to the 1st floor and the husband wants to the 3rd floor. The both hop
on an escalator. The escalators moves them both along at 2 m/s.
Both escalators make an angle of 45° with the horizontal. Find the
husband’s velocity relative to his wife
vhusband
45°
45°
vwife


Position & Displacement Vectors
• The position vector is
defined as:
r
r  x xˆ  y yˆ
• The displacement vector
is defined as:
r r
r
 r  r f  ri

Average Velocity Vector
r
r
• Since  r is a vector, v av must
r
also be a vector as it is  r
times the scalar (1/Δt)
– It is in the same direction
r 
as  r

r
– It will be shorter than  r
as long as t > 1 s
r
r
r
v av 
  t
Example 4.5-4
r
A dragonfly is observed initially at the position r i  2.00 m  xˆ  3.50 m  yˆ
r
Three seconds later it is at the position r f   3 .00 m  xˆ  5 .50 m  yˆ
What is the dragonfly’s displacement and average velocity
during this time?


Example 4.5-4
Find the speed and direction of motion for a rainbow trout whose
r
velocity is v  3.7 m/s  xˆ  1.3 m/s  yˆ


Average Acceleration Vector
r
recall: a av 
r
v
t
maybe more
useful…

r
a av 
r
r
v f  vi
t
Example 4-10 (1D acceleration)
A car is traveling at 25 m/s in a direction 25° E of S
when its driver hits the brakes. It comes to a stop
0.34 s later.
a) Find the car’s change in velocity in unit vector
notation and mag/dir notation.
b) Find the car’s acceleration in unit vector
notation and mag/dir notation.
c) Find the car’s velocity 0.17 s after the driver hits
the brakes. (in unit vector and mag/dir
notation)
Example 4-10 (1D acceleration)
Instantaneous v and a vectors
• If an object’s path is drawn, the velocity vector v is
always in the direction of the object’s motion and
will appear to be tangent to the object’s path
• The acceleration vector a can point in directions
other that the direction of motion, and in general
it does
– a is ALWAYS in the direction of Δv
• The angle between the a and v vectors will
determine if the speed and/or the direction of
motion will change (summarized on the next slide)
How the relative directions of v and a
affect an object’s velocity
Angle between
v&a
Effect on speed
Effect on
direction of
motion
θ = 0°
(same direction)
0° < θ < 90°
speed up
stays same
speed up
turns toward a
θ = 90°
remains constant
turns toward a
90° < θ < 180°
slows down
turns toward a
θ = 180°
(opposite direction)
slows down
stays the same

Example 4-10 (2D acceleration)
Consider a car traveling in a circular path at a
constant speed of 12 m/s. The car if it takes 10 s to
complete ¼ of a revolution.
Calculate the car’s average acceleration (in unit
vector and mag/dir notation) for the portion
of
r
r
r f that
 7 6.4
r ithe
 76car’s
.4 m motion
yˆ
is m
a¼
xˆ revolution from
to

Example 4-10 (2D acceleration)
What is the effect on the object’s speed and direction of motion at points 1, 2, 3, and 4?
2D Kinematic Equations
• We can recycle the 1D kinematic equations, we just
have to be sure to separate all vectors (r, vi, vf, a)
into their components
• Our overall technique will be to TREAT EACH
DIRECTION INDEPENENTLY!
 x  v ix t 
1
2
ax t
2
 y  v iy t 
1
2
ay t
v x  v ix  a x t
v y  v iy  a y t
v v
v v
2
x
2
ix
 2 axx
2
y
2
iy
 2 ayx
2
2D Motion w/ Constant Velocity
Consider a turtle moving over a surface. The turtle
moves with a speed of 5.0 m/s in a direction of 25°
above the +x axis. How far has the turtle moved in
the x and y directions after 5 sec?
Example 4-12
A boat will travel at a speed of 6.5 m/s in still water. It aims
directly across the river. The river’s current is 4.5 m/s
west.
a) What is the boat’s velocity (in mag/dir notation) relative
to the shore?
b) How long did it take the boat to cross the river the river
is 1.25 km wide?
c) How far downstream is the boat when it reaches the
other side?
d) In what direction should the boat aim so that it heads
directly across the river?
e) What would be the boat’s speed while traveling directly
across the river?
Example 4-13
A plane whose airspeed is 200 km/h heads due north
toward its destination. But a 100 km/h northeast
wind (that is, coming from the NE, pointing toward
SW) suddenly begins to blow.
a) What the the resulting velocity of the plane
with respect to the ground?
c) What direction should the pilot fly the plane to
reach his destination?
(d) How long would it take the pilot, assuming he
corrected his course, to reach his destination?
Example #-#
A ball is rolled across a flat table with an initial
velocity of 3.0 m/s North. A fan blows air to the east
that causes the ball to accelerate at 0.4 m/s2 in that
direction.
a.) Find the position of the ball after 4.5 sec.
b.) Find the speed of the ball after 10 sec.
Example #-#
A skier is accelerating down a 30° hill at 3.80 m/s2.
Assume she starts from rest and accelerates
uniformly. Her elevation change is 250 m.
a.) What was the skier’s total displacement?
b.) How long will it take her to reach the bottom of the hill?
c.) What is the skier’s velocity at the bottom of the hill?
(expressed in unit vector notation)
d.) What is the skier’s speed at the bottom of the hill?