Transcript Chapter 2 Motion Along a Straight Line Position

Topic 1.3 Extended
B - Components of motion
Motion in Two Dimensions
3-1 Components of motion
y
y
FYI: The displacement vector gives
Up to now
we have
considered
objects moving in one
the direction
of the
motion
dimension. However, most objects move in more than one
We dimension.
can sketch in our x and y for
successive
snapshots
to obtain
an idea
ofhere:
For example,
consider
the ball
shown
the different velocities the ball has at
different times:
x is in YELLOW.
y is in RED.
We can also sketch in the displacement d of
the ball at each time interval (in GREEN).
Let's examine one time interval in detail:

x
Topic 1.3 Extended
B - Components of motion
x
displacement triangle
vy
If we know the time interval
t between snapshots, we can
find the velocity of the ball
simply by dividing the
displacements shown above by
t. The proportions of our
triangle will not change.
Thus
Magnitude of a
v = vx2 + vy2
2D velocity
y

Each triangle gets a good name:
vy
we can find the value of d if
we know x and y:
d2 = x2 + y2
Magnitude of a
d = x2 + y2
2D displacement
y
From the Pythagorean Theorem

vx
velocity triangle
sin θ =
opposite
θ
adjacent
opp vy
hyp v
cos θ =
vy
component of the velocity.
We call the vy the vertical
component of the velocity.
From trigonometry we know
there is a relationship
between the sides of a
triangle, and the angle :
vy
We call the vx the horizontal

vx
horizontal
component
adj vx
opp vy
tan θ =
hyp v
adj vx
vy = v sin θ
vx = v cos θ
s-o-h-c-a-h-t-o-a
trigonometric ratios
vertical
component
Topic 1.3 Extended
B - Components of motion
vx = (25.0 m/s)cos 30°
vy
of the ball is 25.0 m/s at an
angle of 30° with respect to
(wrt) the positive x-axis.
What is vx the horizontal
component of the velocity?
vx = v cos θ
vy
Suppose we know the velocity

vx
vx = v cos θ
vx = 21.7 m/s
What is vy the vertical
component of the velocity?
vy = v sin θ
vy = (25.0 m/s)sin 30°
vy = 12.5 m/s
FYI: You can check your results by squaring each answer, summing,
and taking the square root. What should you get?
vy = v sin θ
Topic 1.3 Extended
B - Components of motion
Topic 1.3 Extended
B - Components of motion
vy
components of the velocity,
and want to find the magnitude
and the direction:
Suppose vx = 30.0 m/s.
Suppose vy = 40.0 m/s.
Then
v =
vx2 + vy2
v =
302 + 402
v =
50.0 m/s
and
opp
tan θ =
adj
vy
=
vx
θ =
4
3

vx
magnitude
of v
=
40 m/s
30 m/s
so that
tan-1
vy
Sometimes we know the
= 53.1°
direction
of v
Topic 1.3 Extended
B - Components of motion
Sometimes we know the formulas for the components
of the velocity of a ball, and want to find the
magnitude and the direction of the velocity at a
particular time:
Suppose vx = 30.0 (measured in m/s).
Suppose vy = 40.0 - 5t (vy in m/s, t in s)
Then what is the velocity at t = 2 s?
vx = 30.0 m/s
v = vx2 + vy2
vy = 40 - 5(2)
v =
302 + 302
vy = 30.0 m/s
v =
42.4 m/s
magnitude
of v
What is the direction of the ball at this instant?
vy
30 m/s
opp
tan θ =
so that
adj
=
vx
=
30 m/s
θ = tan-1(1) = 45.0°
direction
of v