Transcript presentation name - broward.k12.fl.us

Physics Subject
Area Test
MECHANICS:
KINEMATICS
*
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To simplify the concept of motion,
we will first consider motion that
takes place in one direction.
One example is the motion of a
commuter train on a straight track.
To measure motion, you must first
choose a frame of reference. A
frame of reference is a system for
specifying the precise location of
objects in space and time.
In the train example, any station
along the route.
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Displacement is a change in position.
Displacement is not always equal to the
distance traveled.
The SI unit of displacement is the meter, m.
∆ x = xf -xi
Displacement – final position – initial position
Displacement is not always equal to the
distance traveled.
Example: If a gecko starts at an initial
position of 20 cm and moves to the 80 cm
mark, then retreats back to the 50 cm mark
as its final position, How far has the gecko
traveled? What is its displacement?
The gecko traveled 90 cm, but its
displacement is 30 cm.

In general, right (east) is positive as well as upward
(north) and left (west) is negative as well as
downward (south).
Average velocity is the total displacement divided
by the time interval during which the
displacement occurred.
In SI, the unit of velocity is meters per second
abbreviated as m/s.

Consider a trip to a friend’s house 370 km to
the west (negative direction) along a straight
highway. If you left at 10 AM and arrived at 3
PM, what is your average velocity?
This is your average. You did not travel at
74 km/h at every moment.
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Velocity is not the same as speed.
Velocity describes motion with both direction
and a numerical value (magnitude).
Speed has no direction, only magnitude.
Average speed is equal to the total distance
traveled divided by the time interval.
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Consider an object whose position-time graph is
not a straight line, but a curve.
We obtain different average velocities depending
on the time interval. The instantaneous velocity is
the velocity of the object at a specific point in
the object’s path
The instaneous velocity can
be determined by measuring
the slope of the line that is
tangent to that point on the
diatance-vs-time graph.
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Acceleration – Rate at which velocity changes
over time
An object accelerates if its speed, direction or
both change.
Acceleration has direction and magnitude.
Acceleration is a vector quantity.
Acceleration has the dimensions of
length divided by time squared.
 SI units are m/s2
 Remember we have (m/s)/s = m/s2
Acceleration
A bus slows down with an average acceleration
of -1.8 m/s2. How long does it take the bus to
slow down from 9.0 m/s to a complete stop?
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Consider a train moving to the right, so that the
displacement and velocity are positive.
The slope of the velocity-time graph is the average
acceleration.
• When the velocity in the
positive direction is increasing,
the acceleration is positive, as
at A.
•When the velocity is constant,
there is no acceleration, as at B.
•When the velocity in the
positive direction is decreasing,
the acceleration is negative, as
at C.
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When velocity changes by the same amount
during each time interval, acceleration is
constant.
The relationships between displacement,
time, velocity, and constant acceleration are
expressed by the equations shown on the next
slide.
These equations apply to any object moving
with constant acceleration. These equations
use the following symbols:

A racing car reaches a speed of 42 m/s. It
begins a uniform negative acceleration , using
its parachute and braking system, and comes
to a complete rest 5.5 s later. Find the
distance that the car travels during braking.
Physical Quantities are of 2
types…
*A scalar is only a magnitude
(length) (Example: Temperature,
time, mass)
*A vector has magnitude and
direction (Example: displacement
= 10 m East, Velocity= 50 mph
west)
*A vector will be symbolized by the
“letter” with an arrow over it. The
arrow indicates direction.
A
*Vectors are equal if they have the
same units, magnitude, and
direction.
*A vector can be moved anywhere
parallel to itself.
* To add vectors they must have the same units.
* Tip-to -tail method put them head to tail and
connect them so you end up with a triangle.
* Parallelogram Method- (put them tail to tail)
make vectors parallel and draw a line making 2
triangles
Adding Vectors
(attach)
* Resultant Vector
*The resultant vector is the sum of
a given set of vectors
More Properties of
Vectors
Subtracting Vectors
* Tip to tail- subtract by
putting vector in the
opposite direction
* If you change the sign of a
vector it is not the same
vector. It is a new vector.
* A – B does not equal
B-A
Components of a Vector
* A component is a part
* It is useful to use
rectangular components
* These are the
projections of the
vector along the x- and
y-axes
*
* The x-component of a vector is the projection along the
x-axis
Ax  A cos 
* The y-component of a vector is the projection along the
y-axis
* Then,
Ay  A sin 
A  Ax  Ay
*
A
2
x
2
y
A A
and
 Ay 
  tan  
 Ax 
1
The Pythagorean Theorem can only be used
with right triangles!
When its not 900
R2= A2 + B2 – 2AB(COSӨ)
* Find the magnitude of the sum of a 15 km
displacement and a 25 km displacement when
the angle between them is 900 and when the
angle between them is 1350.
*
(a) Find the horizontal and vertical
components of the 100m
displacement of a superhero
who flies from the top of a tall
building at an angle of 30.00
(b) (b) Suppose instead the
superhero leaps in the other
direction along a displacement
vector B to the top of a flagpole
where the displacement
components are given Bx = -25m
and BY=10.0m. Find the
magnitude and direction of the
displacement vector.
* A GPS receiver indicates that your home is 15.0
km and 400 north of west, but the only path
through the woods leads directly north. If you
follow the path 5.0 km before it opens into a
field, how far, and in what direction, would
you have to walk to reach your home?
* R= 12.39
* Ө= 158’
Resolving a Vector Into Components
The horizontal, or
x-component, of A is
found by Ax = A cos .
The vertical, or
y-component, of A is found by
+y
Ay
A

Ax
Ay = A sin .
+x
By the Pythagorean Theorem, Ax2 + Ay2 = A2
Every vector can be resolved using these formulas, such that A is
the magnitude of A, and  is the angle the vector makes with the xaxis.
Each component must have the proper “sign”
according to the quadrant the vector terminates in.
Analytical Method of Vector Addition
1. Find the x- and y-components of each vector.
Ax = A cos  =
Bx = B cos  =
Cx = C cos  =
Ay = A sin  =
By = B sin  =
Cy = C sin  =
Rx =
Ry =
2. Sum the x-components.
This is the x-component of the resultant.
3. Sum the y-components.
This is the y-component of the resultant.
4. Use the Pythagorean Theorem to find the
magnitude of the resultant vector.
Rx2 + Ry2 = R2
* A roller coaster moves 215 ft horizontally
and then rises 130 ft at an angle of 35.00
above the horizontally. Next, it travels 125
ft at an angle of 50.00 below the
horizontal. Find the roller coaster’s
displacement from its starting point to the
end of this movement.
* A quarter back takes the ball from
the line of scrimmage, runs
backwards for 15.0 yards, then runs
sideways parallel to the line of
scrimmage for 15.0 yards. At this
point, he throws a 60.0 yard forward
pass straight downfield,
perpendicular to the line of
scrimmage. What is the magnitude of
the football’s resultant displacement?
Vector Multiplication
DOT PRODUCT scalar product
A∙B
A ∙ B = AB cosφ
The product of the 2 vectors and the cosine of the angle
between them
A ∙ B = (Axi + Ayj) (Bxi + Byj)
= AxBx i ∙ i + AxBy i ∙ j + AyBx j ∙ i + AyBy j ∙ j
i.i = j.j = k.k = 1
and i.j = j.i = i.k = k.i = j.k = k.j = 0
A ∙ B = AxBx i ∙ i + AyBy j ∙ j
With 3 dimension:
A ∙ B = AxBx i ∙ i + AyBy j ∙ j + AzBz k ∙ k
CROSS PRODUCT
AxB
vector product
The product of the 2 vectors and the sine of the
angle between them
A x B is not the same as B x A
… the direction is opposite
ixi=jxj=kxk=0
ixj=k jxk=I kxi=j
a x b = (a2b3 – a3b2 ) i + (a3b1 – a1b3 ) j + (a1b2- a2b1 ) k
k
i
j
* Two vectors in component forms are written as :
In evaluating the product, we make use of the fact that
multiplication of the same unit vectors gives the value of
0, while multiplication of two different unit vectors result
in remaining vector with appropriate sign. Finally, the
vector product evaluates to vector terms :
*Moving in the x and y direction
*A projectile is an object shot
through the air. This occurs in a
parabola curve.
projectile- any object that moves through the air
or through space, acted on only by gravity
(and air resistance, if any)
Object
dropped
Object
thrown up
Object thrown at an
angle
The vertical acceleration of a
projectile is caused by gravity, so
ay = -9.8 m/s2
Parabolic
Trajectory
* g remains constant (g= -9.8m/s2)
* a in the x direction is 0 because gravity is not
acting on it.
* Neglect air resistance
* Neglect the effects of the earths rotation
Projectiles launched horizontally
To find how far the ball falls, you
use the formula. y =viyt + 1/2gt2
1st second- 5m
After 2 seconds- 20m
After 3 seconds- 45m
The curved path of a
projectile produced is a
parabola (caused by both
horizontal motion and vertical
motion. It must accelerate
only in the vertical direction)
* The projectile will experience two:
* Accelerations (ax= o and aY= -9.8m/s2)
* Velocities
* Displacements
*
Upwardly Launched Projectiles
When a projectile is launched at an upward
angle, it follows a curved path and finally
hits the ground because of gravity.
The Vertical distance a cannonball falls below “imaginary
path if no gravity” is the same vertical distance it
would fall if it were dropped from rest & had been
falling for the same amount of time.
* Draw a free body diagram with a coordinate
system.
* Divide the information into x and y components
* Look at your formulas and decided which
one(s) to use.
Formulas for horizontal and vertical
motion of a projectile
(X) Horizontal
xf-xi = vixt + ½ axt 2
vfx = vix + axt
vfx2 - vix2 = 2ax(xf-xi)
(Y) Vertical
yf-yi = viyt + ½ ayt 2
vfy = viy + ayt
vfy2 = viy2 + 2ay (yf-yi)
Objects that have been thrown will have a
horizontal velocity that stays the same (no
horizontal acceleration ax = 0m/s2)
So vfx =vix in the second formula and third
formulas under horizontal motion.
*
This equation only works when yf and yi are both
the same magnitude
a = 2viy
t
If a ball is thrown up in the air from a moving truck, where will it land?
(Ignore air resistance)
In front of the truck, behind the truck, or back in the truck
Where will a package land if it is released
from a plane?
Behind the plane, in front of the plane
below the plane
What is the horizontal distance covered by
an arrow that was shot through the air at a
600 angle with a velocity of 55 m/s?
Given
Solve
v = 55m/s
Vxi = cos 60(55m/s)=27.5 m/s
vix=27.5 m/s
Viy = sin60(55m/s)=47.6m/s
viy=47.6m/s
To find x dist: x = v0x t
ax=0
Need to find time first!
ay=-9.8m/s2
vfy=viy +ayt
t=?
0 = 47.6m/s + -9.8m/s2 t
dx=?
55m/s
dy
600
Vix
dx
dx = Vix t (we need time)
dx = 27.5 m/s(9.7s)
dx = 266.75m
-47.6m/s = -9.8m/st
4.86 s =t
Total time in the air 4.86s x 2 = 9.7s
* Frames of Reference
Observers using different frames of reference
may measure different displacements or
velocities for an object in motion
Relative Velocities
the difference between the velocities relative to
some common point
*Relative Motion: Suppose you are on a train platform as
the train rushes through the station without stopping.
Someone on board the train is pitching a ball, throwing it
has hard as they can towards the back of the train. If the
train’s speed is 60 mph and the pitcher is capable of
throwing at 60 mph, what is the speed of the ball as you
see it from the platform?
Moving frame of reference
*
A boat heading due north
crosses a river with a
speed of 10.0 km/h. The
water in the river has a
speed of 5.0 km/h due
east.
In general we have vPA
 vPB  vBA
(a) Determine the velocity of the boat.
(b) If the river is 3.0 km wide how long does it take to cross it?
Conservation of Linear Momentum
𝑝1𝑖 + 𝑝2𝑖 = 𝑝1𝑓 + 𝑝2𝑓
Completely Inelastic Collision
𝑚1 𝑣𝑖 = 𝑚1 + 𝑚2 𝑣
𝑚1
𝑣=
𝑣
𝑚1 + 𝑚2 𝑖
Velocity of Center of Mass
𝑝 = 𝑀𝑣𝑐𝑜𝑚 = 𝑚1 + 𝑚2 𝑣𝑐𝑜𝑚
𝑣𝑐𝑜𝑚
𝑝
𝑝1𝑖 + 𝑝2𝑖
=
=
𝑚1 + 𝑚2 𝑚1 + 𝑚2