Download vectors and motion

Vectors and Direction
• In drawing a vector as an
arrow you must choose a
scale.
• If you walk five meters
east, your displacement
can be represented by a 5
cm arrow pointing to the
east.
Vectors and Direction
• Suppose you walk 5 meters east,
turn, go 8 meters north, then
turn and go 3 meters west.
• Your position is now 8 meters
north and 2 meters east of
where you started.
• The diagonal vector that
connects the starting position
with the final position is called
the resultant.
Vectors and Direction
• The resultant is the sum of two
or more vectors added together.
• You could have walked a shorter
distance by going 2 m east and 8
m north, and still ended up in
the same place.
• The resultant shows the most
direct line between the starting
position and the final position.
Calculate a resultant vector
• An ant walks 2 meters West, 3 meters North,
and 6 meters East.
• What is the displacement of the ant?
7.1 Finding Vector Components
Graphically
• Draw a
displacement
vector as an arrow
of appropriate
length at the
specified angle.
• Mark the angle and
use a ruler to draw
the arrow.
7.1 Finding the Magnitude of a Vector
• When you know the x- and y- components of a vector, and
the vectors form a right triangle, you can find the magnitude
using the Pythagorean theorem.
7.1 Adding Vectors
• Writing vectors in components make it easy to add them.
7.1 Subtracting Vectors
7.1 Calculate vector magnitude
• A mail-delivery robot
needs to get from where it
is to the mail bin on the
map.
• Find a sequence of two
displacement vectors that
will allow the robot to
avoid hitting the desk in
the middle.
7.2 Projectile Motion and the Velocity
Vector
• Any object that is moving
through the air affected
only by gravity is called a
projectile.
• The path a projectile
follows is called its
trajectory.
7.2 Projectile Motion and the Velocity
Vector
• The trajectory of a
thrown basketball follows
a special type of archshaped curve called a
parabola.
• The distance a projectile
travels horizontally is
called its range.
7.2 Projectile Motion and the Velocity
Vector
• The velocity vector (v) is a
way to precisely describe the
speed and direction of
motion.
• There are two ways to
represent velocity.
• Both tell how fast and in what
direction the ball travels.
7.2 Calculate magnitude
Draw the velocity vector v =
(5, 5) m/sec and calculate
the magnitude of the
velocity (the speed), using
the Pythagorean theorem.
7.2 Components of the Velocity Vector
• Suppose a car is driving 20
meters per second.
• The direction of the vector
is 127 degrees.
• The polar representation of
the velocity is v = (20
m/sec, 127°).
7.2 Calculate velocity
• A soccer ball is kicked at a speed of 10 m/s and an angle of
30 degrees.
• Find the horizontal and vertical components of the ball’s
initial velocity.
7.2 Adding Velocity Components
• Sometimes the total velocity of an object is a combination of
velocities.
 One example is the motion of a boat on a river.
 The boat moves with a certain velocity relative to the
water.
 The water is also moving with another velocity relative to
the land.
7.2 Adding Velocity Components
7.2 Calculate velocity components
• An airplane is moving at a velocity of 100 m/s in a direction 30
degrees NE relative to the air.
• The wind is blowing 40 m/s in a direction 45 degrees SE relative to
the ground.
• Find the resultant velocity of the airplane relative to the ground.
7.2 Projectile Motion
Vx
• When we drop a ball
from a height we know
that its speed increases
as it falls.
• The increase in speed is
due to the acceleration
gravity, g = 9.8 m/sec2.
Vy
y
x
7.2 Horizontal Speed
• The ball’s horizontal velocity
remains constant while it falls
because gravity does not exert
any horizontal force.
• Since there is no force, the
horizontal acceleration is zero
(ax = 0).
• The ball will keep moving to
the right at 5 m/sec.
7.2 Horizontal Speed
• The horizontal distance a projectile moves can be
calculated according to the formula:
7.2 Vertical Speed
• The vertical speed (vy) of the
ball will increase by 9.8 m/sec
after each second.
• After one second has passed, vy
of the ball will be 9.8 m/sec.
• After the 2nd second has
passed, vy will be 19.6 m/sec
and so on.
7.2 Calculate using projectile motion
• A stunt driver steers a car off
a cliff at a speed of 20 meters
per second.
• He lands in the lake below
two seconds later.
• Find the height of the cliff
and the horizontal distance
the car travels.
7.2 Projectiles Launched at an Angle
• A soccer ball kicked
off the ground is also
a projectile, but it
starts with an initial
velocity that has
both vertical and
horizontal
components.
*The launch angle determines how the initial velocity
divides between vertical (y) and horizontal (x) directions.
7.2 Steep Angle
• A ball launched at
a steep angle will
have a large
vertical velocity
component and a
small horizontal
velocity.
7.2 Shallow Angle
• A ball launched at a
low angle will have
a large horizontal
velocity component
and a small vertical
one.
7.2 Projectiles Launched at an Angle
The initial velocity components of an object launched at a velocity vo
and angle θ are found by breaking the velocity into x and y
components.
7.2 Range of a Projectile
• The range, or horizontal distance, traveled by a projectile
depends on the launch speed and the launch angle.
7.2 Range of a Projectile
• The range of a projectile is calculated from the
horizontal velocity and the time of flight.
7.2 Range of a Projectile
• A projectile travels farthest when launched at 45
degrees.
7.2 Range of a Projectile
• The vertical velocity is responsible for giving the
projectile its "hang" time.
7.2 "Hang Time"
•
•
•
•
•
You can easily calculate your own hang time.
Run toward a doorway and jump as high as you can, touching the wall or door frame.
Have someone watch to see exactly how high you reach.
Measure this distance with a meter stick.
The vertical distance formula can be rearranged to solve for time:
7.2 Projectile Motion and the Velocity
Vector
Key Question:
Can you predict the landing spot of a projectile?
*Students read Section 7.2 BEFORE Investigation 7.2
Marble’s Path
Vx
t=?
Vy
y
x=?
In order to solve “x” we must know “t”
Y = vot – ½ g t2
vot = 0 (zero)
Y = ½ g t2
2y = g t2
t2 = 2y
g
t = 2y
g
7.3 Forces in Two Dimensions
• Force is also represented in x-y components.
7.3 Force Vectors
• If an object is in equilibrium,
all of the forces acting on it
are balanced and the net
force is zero.
• If the forces act in two
dimensions, then all of the
forces in the x-direction and
y-direction balance
separately.
7.3 Equilibrium and Forces
• It is much more difficult
for a gymnast to hold his
arms out at a 45-degree
angle.
• To see why, consider that
each arm must still
support 350 newtons
vertically to balance the
force of gravity.
7.3 Forces in Two Dimensions
• Use the y-component to find the total force in the gymnast’s
left arm.
7.3 Forces in Two Dimensions
• The force in the right arm must also be 495 newtons because
it also has a vertical component of 350 N.
7.3 Forces in Two Dimensions
• When the gymnast’s arms
are at an angle, only part of
the force from each arm is
vertical.
• The total force must be
larger because the vertical
component of force in each
arm must still equal half his
weight.
7.3 Forces and Inclined Planes
• An inclined plane is a straight surface, usually with a
slope.
• Consider a block sliding down
a ramp.
• There are three forces that
act on the block:
– gravity (weight).
– friction
– the reaction force acting
on the block.
7.3 Forces and Inclined Planes
• When discussing forces, the word “normal” means
“perpendicular to.”
• The normal force acting
on the block is the
reaction force from the
weight of the block
pressing against the
ramp.
7.3 Forces and Inclined Planes
• The normal force
on the block is
equal and opposite
to the component
of the block’s
weight
perpendicular to
the ramp (Fy).
7.3 Forces and Inclined Planes
• The force parallel to
the surface (Fx) is
given by
Fx = mg sinθ.
7.3 Acceleration on a Ramp
• Newton’s second law can be used to calculate the
acceleration once you know the components of all the forces
on an incline.
• According to the second law:
Acceleration
(m/sec2)
a=F
m
Force (kg . m/sec2)
Mass (kg)
7.3 Acceleration on a Ramp
• Since the block can only accelerate along the ramp, the force that
matters is the net force in the x direction, parallel to the ramp.
• If we ignore friction, and substitute Newtons' 2nd Law, the net force
is:
Fx = m g sin θ
a= F
m
7.3 Acceleration on a Ramp
• To account for friction, the horizontal component of
acceleration is reduced by combining equations:
Fx = mg sin θ - m mg cos θ
7.3 Acceleration on a Ramp
• For a smooth surface, the coefficient of friction (μ) is usually in
the range 0.1 - 0.3.
• The resulting equation for acceleration is:
7.3 Calculate acceleration on a ramp
• A skier with a mass of 50 kg is on a hill making an angle of 20
degrees.
• The friction force is 30 N.
• What is the skier’s acceleration?
7.3 Vectors and Direction
Key Question:
How do forces balance in
two dimensions?
*Students read Section 7.3 BEFORE Investigation 7.3
Application: Robot Navigation