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Ch 3: Motion in 2-D
Everything we have learned is wrong!
Objects will not keep accelerating until they
reach the ground from large heights.
Terminal velocity – the maximum speed an object
can obtain while free falling through a fluid (air)
Ex: Throw a penny off the Empire state building
Height: 380 m
Velocity of penny at bottom = 86 m/s = 192 mph!!!!
Terminal velocity of penny: 30-50 mph depending
on updraft
Trig Functions to Find Vector Components
We can use all of this
to add vectors analytically!
3-2 Addition of Vectors—Graphical Methods
Resultant = what you get when you
add/subtract vectors
For vectors in one
dimension, simple addition
and subtraction are all that
is needed.
[Pythagorean Theorem]
You do need to be careful
about the signs, as the
figure indicates.
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3-2 Addition of Vectors—Graphical Methods
If the motion is in two dimensions, the situation is somewhat
more complicated.
Example
A car drives east 10 miles, turns left and drives 4 miles north,
what is the angle and magnitude of the resultant vector?
Here, the actual travel paths are at right angles to one another;
we can find the displacement by using the Pythagorean
Theorem.
3-2 Addition of Vectors—Graphical Methods
Even if the vectors are not at right angles,
they can be added graphically by using the
tail-to-tip method.
3-2 Addition of Vectors—Graphical Methods
The parallelogram method may also be used; here
again the vectors must be tail-to-tip.
Drawing out the addition of the vectors and
the resultant helps visualize whats going on
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3-4 Adding Vectors by Components
3-4 Adding Vectors by Components
The components are effectively one-dimensional, so
they can be added arithmetically.
If the components are
perpendicular, they can be
found using trigonometric
functions.
Example
A rural mail carrier leaves the post office & drives
22.0 km in a northerly direction. She then drives
in a direction 60.0° south of east for 47.0 km.
What is her displacement from the post office?
3-4 Adding Vectors by Components
Adding vectors:
1. Draw a diagram; add the vectors graphically.
2. Choose x and y axes.
3. Resolve each vector into x and y components.
4. Calculate each component using sines and cosines.
5. Add the components in each direction.
6. To find the length and direction of the vector, use:
.
and
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3-4 Adding Vectors by
Components
3-9 Relative Velocity
Example 3-14: Heading upstream.
Example 3-3: Three short trips.
An airplane trip involves three
legs, with two stopovers. The first
leg is due east for 620 km; the
second leg is southeast (45°) for
440 km; and the third leg is at 53°
south of west, for 550 km, as
shown. What is the plane’s total
displacement?
3-9 Relative Velocity
Example 3-15: Heading across the river.
The same boat (vBW = 1.85 m/s)
now heads directly across the
river whose current is still 1.20
m/s. (a) What is the velocity
(magnitude and direction) of the
boat relative to the shore? (b) If
the river is 110 m wide, how long
will it take to cross and how far
downstream will the boat be
then?
A boat’s speed in still
water is vBW = 1.85 m/s. If
the boat is to travel
directly across a river
whose current has speed
vWS = 1.20 m/s, at what
upstream angle must the
boat head?
3-5 Unit Vectors
Unit vectors have magnitude 1.
Using unit vectors, any vector
can be written in terms of its
components:
r = (rxi + ryj + rzk)
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Example:
3-6 Vector Kinematics
The same definitions for velocity and
acceleration apply when using unit vectors
The position of a particle is given by
r = (9.6ti + 8.85j - 1.00t2k) m
Determine the particle’s velocity and acceleration as
a function of time.
Note this is a particle and not a projectile.
Section 3-7: Projectile Motion
A projectile is an
object moving in two
dimensions under
the influence of
Earth's gravity; its
path is a parabola.
Projectile Motion ≡ Motion of an object that
is projected into the air at an angle.
1) Components of motion can
be treated separately
2) There is NO acceleration
in the x direction
3) y direction is just a
freefall problem
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Ball Rolls Across Table & Falls Off
t = 0 here
↓
Take down as positive. Initial
velocity has an x component
ONLY! That is vy0 = 0.
At any point, v has both x &
y components. Kinematic
equations tell us that, at time
t,
vx = vx0
vy = gt
x = vx0t
y = vy0t + ½gt2
Example 3-6: Driving off a cliff
A movie stunt driver on a motorcycle speeds horizontally off a 50.0-m-high
cliff. How fast must the motorcycle leave the cliff top to land on level
ground below, 90.0 m from the base of the cliff where the cameras are?
What is it asking?
Example 3-6: Driving off a cliff!!
A movie stunt driver on a motorcycle speeds horizontally off a 50.0-m-high
cliff. How fast must the motorcycle leave the cliff top to land on level
ground below, 90.0 m from the base of the cliff where the cameras are?
What is it asking?
vo=?
What do we know?
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Example 3-6: Driving off a cliff!!
A movie stunt driver on a motorcycle speeds horizontally off a 50.0-m-high
cliff. How fast must the motorcycle leave the cliff top to land on level
ground below, 90.0 m from the base of the cliff where the cameras are?
Example 3-6: Driving off a cliff!!
A movie stunt driver on a motorcycle speeds horizontally off a 50.0-m-high
cliff. How fast must the motorcycle leave the cliff top to land on level
ground below, 90.0 m from the base of the cliff where the cameras are?
What is it asking?
vo=?
What do we know?
x = 90 m
y = 50 m
Also
y is positive upward
xo = y0 = 0 at top.
vy0 = 0
Solving Problems Involving Projectile Motion
Example 3-7: Kicked football
1. Read the problem carefully, & choose the object(s) you
are going to analyze.
2. Sketch a diagram.
3. Choose an origin & a coordinate system.
4. Decompose vectors if needed.
5. Solve for the x and y motions separately.
6. List known & unknown quantities. Remember that vx never
changes, & that vy = 0 at the highest point.
A football is kicked at an angle θ0 = 37.0° with a velocity of 20.0 m/s, as
shown. Calculate:
a. Max height. b. Time when hits ground. c. Total distance
traveled in the x direction. d. Velocity at top. e. Acceleration at
top.
7. Plan how you will proceed. Use the appropriate equations;
you may have to combine some of them.
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Example 3-7: Kicked football
Example 3-7: Kicked football
First: Resolve vo into its x and y components.
A football is kicked at an angle θ0 = 37.0° with a velocity of 20.0 m/s,
as shown. Calculate:
vox = vo cos ϴ = 20 m/s cos(37o) = 16 m/s
a. Max height.
voy = vo sin ϴ = 20 m/s
sin(37o)
= 12 m/s
Example 3-7: Kicked football
A football is kicked at an angle θ0 = 37.0° with a velocity of 20.0 m/s,
as shown. Calculate:
b. Time when it hits the ground.
A
B
Which equation?
A
B
C
D
E
Example 3-6: Kicked football
A football is kicked at an angle θ0 = 37.0° with a velocity of 20.0 m/s,
as shown. Calculate:
c. Total distance traveled in the x direction.
C
D
E
A
B
C
D
E
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Example 3-7: Kicked football
Example 3-7: Kicked football
A football is kicked at an angle θ0 = 37.0° with a velocity of 20.0 m/s,
as shown. Calculate:
A football is kicked at an angle θ0 = 37.0° with a velocity of 20.0 m/s,
as shown. Calculate:
d. Total velocity at top.
A) 16 m/s
D) 0 m/s
e. Acceleration at top.
A) 9.8 m/s2
B) 8 m/s
B) 10 m/s2
C) 12 m/s
C) 4.9 m/s2
D) 0 m/s2
Keep in Mind
Example: A punt
Choose an origin & a coordinate system.
Suppose the football in Example 3–7 was punted
and left the punter’s foot at a height of 1.00 m
above the ground. How far did the football travel
before hitting the ground? Set x0 = 0, y0 = 0.
Decompose vectors if needed.
Solve for the x and y motions separately.
There is NO acceleration in X direction!
At the top of the arc vy = 0
You may have to use TWO equations to get an answer.
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