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C. Relative Velocity
The speed of an object is always in reference to some other
object.
For example, when we say an car is moving at 50 km/h,
this is in reference to the ground. The ground is assumed to
be at rest.
50 km/h
ground (at rest)
But what if the ground (i.e. the reference frame) is moving?
50 km/h
ground
10 km/h
In order to determine the velocity of an object relative to
a moving reference frame, we need to use vector arithmetic.
Consider a person walking on a train:
4.0 m/s 10.0 m/s
The person is walking 4.0 m/s East (relative to the train),
while the train is moving 10.0 m/s (relative to the ground).
How fast is the person moving relative to the ground?
4.0 m/s 10.0 m/s
Since the person and the train are moving in the same direction,
the person appears to be moving even faster than the train.
vperson = (+4.0 m/s) + (+10.0 m/s) Ref: East +
= +14.0 m/s West -
= 14.0 m/s East
What if the person is walking in the opposite direction?
4.0 m/s 10.0 m/s
The person is walking 4.0 m/s West (relative to the train),
while the train is moving 10.0 m/s East.
How fast is the person moving relative to the ground?
4.0 m/s 10.0 m/s
The person and the train are moving in opposite directions.
But the train is moving faster.
vperson = (-4.0 m/s) + (+10.0 m/s) Ref: East +
= +6.0 m/s West
-
= 6.0 m/s East
Summary (1-D Relative Velocity)
If both objects are moving on the same axis:
State a reference system
- choose a positive and a negative direction
Simply add the vectors to get the relative velocity
Ex. 1 A boat, capable is travelling 9.0 m/s in still water, heads
East on a river with a current moving 4.0 m/s West.
a) What is its resultant velocity?
b) How long would it take to travel 6.0 km upstream?
a) current
4.0 m/s
9.0 m/s
Ref: East +
West - vboat = (+9.0 m/s) + (-4.0 m/s)
= +5.0 m/s
= 5.0 m/s East
C2. 2-D Relative Velocity
We will now consider relative velocity in two dimensions.
This is especially useful for navigation.
Terminology:
Heading = The direction the plane is aimed
i.e. the way it would travel if there is no wind
Airspeed = The speed of the plane relative to the air
i.e. the speed it would travel if there was no wind
Groundspeed = The speed relative to the ground
i.e. the actual speed of the plane
There are two kinds of problems you will deal with:
1. Crosswind (or cross-current)
- the wind will blow you "off course"
2. Heading into the wind (or upstream)
- in order to get to a destination, you must
aim your craft into the wind
Ex. 1 (Crosswind question)
A plane heads directly North with an airspeed of 300 km/h.
However, there is a 50.0 km/h wind blowing from the East
(i.e. towards the West).
a) What is the resultant velocity of the plane?
b) How far would it get blown off-course in 40.0 minutes?
a) N
50.0 km/h wind
R 300 km/h
plane
W E
Since the velocity vectors are at right angles (i.e. 90),add them tail-to-tip.
The resultant R represents the plane's actual velocity
(with respect to the ground).
a) 50.0 km/h
R 300 km/h
Pythag:
c2 = a2 + b2
R2 = (50.0 km/h)2 + (300 km/h)2
R = 304 km/h
Notice that the wind moves the plane even faster.
a) N
304 km/h
9.46
W E
So, the actual velocity of the plane (relative to the ground)
is
304 km/h at 9.46 W of N 80.5 N of W
99.5 rcs
b) First, find how far the plane travelled.
v = d
t
d = v t
= (304.138 km/h) (40 / 60 hours)
= (304.138 km/h) (0.66667 h)
= 202.76 km
b) N
202.76 km
9.46
W E
The plane's displacement is in the same direction
as its actual velocity.
Thus, it is directed at 9.46 W of N.
Ex. 2 (Heading "into the wind")
A plane must fly directly West a distance of 1200 km.
However, there is a 65.0 km/h wind towards the North.
If the plane's airspeed is 340 km/h,
a) what heading is needed?
b) what is the flight time?
a)
Destination
340 km/h
If there was no wind, the plane could aim directly
West and travel at a speed of 340 km/h.
a)
Destination
However, if the plane aims West and there is a wind blowing
towards the North, then it will be blown off-course.
a)
Destination
The plane must aim "into the wind". Then, the wind will blow
the plane "on-course" and it will arrive at its destination.
b) v
11.0 60 km/h
340 km/h
Find actual speed (groundspeed):
c2 = a2 + b2
3402 = v2 + 602
v2 = 3402 - 602
v = 333.73 km/h Notice that the wind slowed
the plane down.
Animations
Boat on Water:
http://mysite.verizon.net/vzeoacw1/velocity_composition.html