Transcript Advanced Physics

Advanced Physics
Chapter 3
Kinematics in Two Dimensions;
Vectors
Chapter 3
3-1 Vectors and Scalars
3-2 Addition of Vectors—Graphical Methods
3-3 Subtraction of Vectors, and Multiplication of a
Vector by a Scalar
3-4 Adding Vectors by Components
3-5 Projectile Motion
3-6 Solving Problems Involving Projectile Motion
3-7 Projectile Motion is Parabolic
3-8 Relative Velocity
3-1 Vectors and Scalars
Scalar
A quantity that has only magnitude (size)
and units
Mass, time, distance, speed
3-1 Vectors and Scalars
Vector
A quantity that has both direction and magnitude.
It also has units!
Displacement, velocity, acceleration, force
When drawing a vector quantity use an arrow
whose length represents its size and points in the
direction of the vector
To represent a vector quantity in an equation, put
an arrow over the letter symbol.
Example: velocity = v
3-2 Addition of Vectors—
Graphical Methods
Two or more vectors that measure the same
quantity can be added together to get their
sum
Resultant—sum of vectors
3-2 Addition of Vectors—
Graphical Methods
To add vectors graphically (The Butt-head Method):
1. Draw first vector to scale with in correct
direction.
2. Draw the second vector with its butt (tail) to the
head of the first vector.
3. Continue to draw additional vectors butt to head.
4. Connect the butt of the first vector with the butt
of a resultant vector and the head of the last
vector to the head of a resultant vector.
5. Measure the size and direction of the resultant
vector; this is the sum of the vectors.
3-2 Addition of Vectors—
Graphical Methods
To add vectors graphically (The Parallelogram
Method):
1. Draw first vector to scale with in correct
direction.
2. Draw the second vector with its butt (tail) to the
butt of the first vector.
3. Draw a parallelogram using these two sides.
4. Draw the resultant vector so that it is diagonal
from the common origin.
5. Measure the size and direction of the resultant
vector; this is the sum of the vectors.
3-2 Addition of Vectors—
Graphical Methods
To add two perpendicular vectors mathematically:
1. Draw first vector in correct direction.
2. Draw the second vector with its butt (tail) to the
head of the first vector.
3. Connect the butt of the first vector with the butt
of a resultant vector and the head of the last
vector to the head of a resultant vector.
4. Find the size of the resultant using the
Pythagorean Theorem.
5. Find the direction of resultant by using SOH,
CAH, TOA trigonometry.
3-2 Addition of Vectors—
Graphical Methods
To add any two vectors mathematically:
1. Draw first vector in correct direction.
2. Draw the second vector with its butt (tail) to the head of
the first vector.
3. Connect the butt of the first vector with the butt of a
resultant vector and the head of the last vector to the
head of a resultant vector.
4. Find the size of the resultant using the Law of Cosines
c2 = a2 + b2 – (2ab cos )
5.
Find the direction of resultant by using the Law of Sines
sin /a = sin /b = sin/c or
a/sin A = b/ sin B = c/ sin C
Important!
3-3 Subtraction of Vectors, and
Multiplication of a Vector by a
Scalar
Subtraction of Vectors
Use the butt-head method but reverse the
direction of the second vector (or any
subtracted)
Multiplication of a Vector by a Scalar
Increases the size of the vector by the
magnitude of the scalar
3-4 Adding Vectors by
Components
A single vector can be thought of as being made
up of two perpendicular vectors.
These perpendicular vectors are called the vector’s
components.
Usually the components are chosen to be along the
x and y axes.
Finding the components of a single vector is called
resolving the vector into its components
3-4 Adding Vectors by
Components
When adding vectors you can either us
trigonometry or you may first find the x and y
components of each vector, add the
components together and then find the
resultant of these two vectors.
This is called adding vectors by components.
3-4 Adding Vectors by
Components
Sample problem:
A mailperson leaves the post office and drives
22km due north and then 47km 60º south of
east. Find her displacement from the post
office. Use BOTH trigonometry and
component addition
3-4 Adding Vectors by
Components
Sample problem:
A mailperson leaves the post office and drives
22km due north and then 47km 60º south of
east. Find her displacement from the post
office.
30 km 38.5ºsouth of east
3-5 Projectile Motion
Projectile Motion
When an object moves through the air only under
the force of gravity.
Air resistance is ignored!
The object has a horizontal and a vertical
component of its motion.
These horizontal and vertical components are
independent of each other.
When dealing with projectile motion each
component is analyzed separately
3-5 Projectile Motion
Horizontal Motion
no net force acts on object so it doesn’t accelerate
Initial velocity (horizontal) = final velocity
(horizontal)
Vertical motion
A net force acts on the object so it accelerates it
downward (g = 9.8 m/s2)
Velocity changes magnitude and/or direction as
the object moves through the air
3-5 Projectile Motion
Since all objects fall at same rate, a ball
dropped or thrown horizontally from the
same height will hit the ground at the same
time!
3-5 Projectile Motion
An object projected at an upward angle follows a
path where:
Horizontal component of velocity is constant
Vertical component of velocity has same
magnitude but different directions at two places.
Vertical component of velocity is zero at top of
path
Acceleration due to gravity acts on vertical
component only and it acts downward.
3-6 Solving Problems Involving
Projectile Motion
1. Read problem and draw a diagram
2. Separate velocity vector into its horizontal and
3.
4.
5.
6.
vertical components.
Write down all vertical and horizontal knowns
and unknowns
Solve for vertical and horizontal components
separately
Find the ½ time using vertical components
Use ½ time to find maximum height
1. (vtop = 0, a = g = – 9.8 m/s2)
7. Use ½ time to find hang time and then range
3-6 Solving Problems Involving
Projectile Motion
Sample Problem:
A football is kicked at an angle of 37º with a
velocity of 20.0 m/s. Find its:
Hang time
Maximum height
Range
3-6 Solving Problems Involving
Projectile Motion
Sample Problem:
A football is kicked at an angle of 37º with a velocity
of 20.0 m/s. Find its:
Hang time
2.45 s
Maximum height
7.35 m
Range
39.2 m
3-7 Projectile Motion is Parabolic
Duh!
3-8 Relative Velocity
If observations are made in different reference
frames objects may appear differently to observers
in each frame.
Example #1
A person in a moving car drops a can out the
window. To the person inside the car the can
seems to fall backwards. To a person standing
outside the car the can seems to fall forward.
3-8 Relative Velocity
Example #2
A boat moves at a speed of 10 m/s straight across a
river. The river flows perpendicularly to the boat
a 5 m/s.
How fast is the boat approaching a person floating
in a lifejacket straight across from the boat.
How fast is the boat moving relative to a person
standing on the shore?
3-8 Relative Velocity
Example #2
A boat moves at a speed of 10 m/s straight across a
river. The river flows perpendicularly to the boat
a 5 m/s.
How fast is the boat approaching a person floating
in a lifejacket straight across from the boat?
10m/s
How fast is the boat moving relative to a person
standing on the shore?
11.2 m/s