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Kinematics in Two Dimensions


x = x0 + v0xt + 1/2 axt2 vx = v0x + axt vx2 = v0x2 + 2ax x y = y0 + v0yt + 1/2 ayt2 vy = v0y + ayt vy2 = v0y2 + 2ay y Kinematics for Projectile Motion – PowerPoint PPT presentation

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Title: Kinematics in Two Dimensions

Kinematics in Two Dimensions
  • x x0 v0xt 1/2 axt2
  • vx v0x axt
  • vx2 v0x2 2ax ?x
  • y y0 v0yt 1/2 ayt2
  • vy v0y ayt
  • vy2 v0y2 2ay ?y

Kinematics for Projectile Motion ax 0
ay -g
  • y y0 v0yt - 1/2 gt2
  • vy v0y - gt
  • vy2 v0y2 - 2g ?y
  • x x0 vxt
  • vx v0x

x and y motions are independent! They share a
common time t
Example 1
  • You and a friend are standing on level ground,
    each holding identical baseballs. At exactly the
    same time, and from the same height, you drop
    your baseball without throwing it while your
    friend throws her baseball horizontally as hard
    as she can. Which ball hits the ground first?
  • 1. Your ball
  • 2. Your friends ball
  • 3. They both hit the ground at the same time

They both have the same initial vertical
component with the same acceleration due to
gravity, therefore they hit the ground at the
same time.
No matter how much horizontal velocity is put on
an object it still falls at the same rate as any
other dropped object.
  • y y0 voyt - gt2/2
  • v0y 0 and y0
  • Therefore, tsqrt(2y0/g)
  • Result is independent of v0x

Example 2
A flatbed railroad car is moving along a track at
constant velocity. A passenger at the center of
the car throws a ball straight up. Neglecting
air resistance, where will the ball land ? 1.
Forward of the center of the car 2. At the center
of the car 3. Backward of the center of the car
The train and the ball have the same horizontal
velocity and by throwing the ball straight up,
the horizontal component is not changed.
The ball has no acceleration in the horizontal
direction. Therefore, the balls remains directly
above the center of the train at all times during
the flight and would fall directly back toward
the center of the train.
Example 4
  • You are a vet trying to shoot a tranquilizer dart
    into a monkey hanging from a branch in a distant
    tree. You know that the monkey is very nervous,
    and will let go of the branch and start to fall
    as soon as your gun goes off. On the other hand,
    you also know that the dart will not travel in a
    straight line, but rather in a parabolic path
    like any other projectile. In order to hit the
    monkey with the dart, where should you point the
    gun before shooting?
  • 1 Right at the monkey
  • 2 Below the monkey
  • 3 Above the monkey

If the shot is fired at the monkey the same time
the monkey drops, both objects will fall at the
same rate causing the shot to hit the monkey.
Example 4 Shooting the Monkey... (II)
x v0 t y -1/2 g t2
x x0 y -1/2 g t2
No monkeys were harmed during the making of this
Example 4 Shooting the Monkey... (III)
y y0 - 1/2 g t2
  • At an angle, still aim at the monkey!

y v0 t - 1/2 g t2
Newton's Laws
After this lecture, you should know
about Force, mass, inertia. Newtons first and
second law. Inertial and non-inertial reference
frame. Gravitation. Action reaction. Normal
force. Free body diagram.
Classical Mechanics and forces
  • Classical mechanics
  • Describes the relationship between the motion of
    objects in our everyday world and the forces
    acting on them
  • Conditions when Classical Mechanics does not
  • very tiny objects (lt atomic sizes)
  • objects moving near the speed of light
  • Force
  • Usually think of a force as a push or pull
  • Vector quantity
  • May be a contact force or a field force
  • Contact forces result from physical contact
    between two objects
  • Field forces act between disconnected objects
  • Also called action at a distance

Contact and Field Forces
Fundamental Forces
  • Types
  • Strong nuclear force
  • Electromagnetic force
  • Weak nuclear force
  • Gravity
  • Characteristics
  • All field forces
  • Listed in order of decreasing strength
  • Only gravity and electromagnetic in mechanics

Newtons First Law
  • The motion of an object does not change unless it
    is acted upon by a net external force (for the
    definition of force see Newtons 2nd Law)
  • If v0, it remains 0
  • If v is some value, it stays at that value
  • Hence
  • If no net force
  • velocity is constant in magnitude and direction
  • acceleration is zero
  • Hence An object traveling at a constant velocity
    along a straight line will continue to do so as
    long as there is no net external force acting on
  • External forces are forces that result from the
    interaction between the object and its

Example 7
  • An airplane is flying from Madison to O'Hare.
    Many forces act on the plane, including weight
    (gravity), drag (air resistance), the thrust of
    the engine, and the lift of the wings. At some
    point during its trip the velocity of the plane
    is measured to be constant (which means its
    altitude is also constant). At this time, the
    total force on the plane 1. is pointing
    upward2. is pointing downward 3. is pointing
    forward 4. is pointing backward5. is zero

When the velocity is constant the objects
acceleration is equal to zero. The only time
acceleration is equal to zero is when the sum of
the net force is equal to zero.
An object traveling at a constant velocity along
a straight line will continue to do so as long as
there is no net force acting on it (Newton's
First Law). The total force acting on the plane
is zero, because its motion is uniform in a
straight line.
Inertia and mass
  • Inertia
  • Is the tendency of an object to continue in its
    original motion
  • (N.B. not a physics quantity in the strict
    sense of the term)
  • Mass
  • A measure of the resistance of an object to
    changes in its motion due to a force
  • Scalar quantity
  • SI units are kg

Inertial vs non-inertial reference frame
  • Inertial reference frames are coordinate systems
    which travel at constant velocity.
  • In such a frame, an object is observed to have no
    acceleration when no forces are acting on it.
  • If a reference frame moves with constant velocity
    relative to an inertial reference frame, it also
    is an inertial reference frame.
  • There is no absolute inertial reference frame,
    meaning that there is no state of velocity which
    is special in the universe. All inertial
    reference frames are equivalent. One can only
    detect the relative motion of one inertial
    reference frame to another.
  • (Approximate) example of inertial frame train
    moving with constant velocity
  • Examples of non-inertial frames train
    accelerating, bus braking, car on a curve
  • Newtons 1st law is a way to define inertial

Newtons Second Law
  • The acceleration of an object is directly
    proportional to the net force acting on it and
    inversely proportional to its mass.
  • Units
  • F M a
  • F kg-m/s2
  • 1 Newton (N) 1 kg-m/s2
  • A vector equation
  • Fnet,x Max
  • Fnet,y May
  • Newtons second law applies only in inertial
    reference frames
  • Applying it in non-inertial ones leads to
    pseudo-forces, e.g. centrifugal force

Example 8
  • A force F acting on a mass m1 results in an
    acceleration a1.The same force acting on a
    different mass m2 results in an acceleration a2
    2a1. What is the mass m2?

(1) 2m1 (2) m1 (3) m1/2
  • Fma
  • F m1a1 m2a2 m2(2a1)
  • Therefore, m2 m1/2
  • Or in wordstwice the acceleration means half the

Gravitational Force and Weight
  • Mutual force of attraction between any two
  • Expressed by Newtons Law of Universal
  • The magnitude of the gravitational force acting
    on an object of mass m near the Earths surface
    is called the weight w of the object
  • w m g is a special case of Newtons Second Law
  • g is the acceleration due to gravity g 9.81
  • g can also be found from the Law of Universal
  • g GMearth/r2
  • Weight is not an inherent property of an object
  • mass is an inherent property
  • Weight depends upon location

Weight and Mass
Example 9
What is the approximate weight force of a bar of
Chocolate of 100g on sea level ?
Use g10m/s2 W 0.1kg x 10 m/s2 1 N
1 N
Newtons Third Law
  • For every action, there is an equal and opposite
  • Finger pushes on box
  • Ffinger?box force exerted on box by finger
  • Box pushes on finger
  • Fbox?finger force exerted on finger by box
  • Third Law
  • Fbox?finger - Ffinger?box

Newton's Third Law...
  • FA ,B - FB ,A. is true for all types of

Whenever one body exerts a force on a second
body, the first body experiences a force that is
equal in magnitude and opposite in direction to
the one it exerts.
Example of Bad Thinking
  • Since Fm,b -Fb,m why isnt Fnet 0, and a 0 ?

a ??
Example of Good Thinking
  • Consider only the box!
  • Fon box mabox Fm,b
  • Free Body Diagram (more on this next time)

What about forces on man?
Example 10A
  • Suppose you are an astronaut in outer space
    giving a brief push to a spacecraft whose mass is
    bigger than your own.
  • 1) Compare the magnitude of the force you exert
    on the spacecraft, FS, to the magnitude of the
    force exerted by the spacecraft on you, FA, while
    you are pushing1. FA FS 2. FA gt FS3. FA
    lt FS

Third Law!
Example 10B
2) Compare the magnitudes of the acceleration
you experience, aA, to the magnitude of the
acceleration of the spacecraft, aS, while you
are pushing 1. aA aS 2. aA gt aS 3. aA lt aS
aF/m F same, hence lower mass gives larger a
Example 11
Consider a car at rest. We can conclude that the
downward gravitational pull of Earth on the car
and the upward contact force of Earth on it are
equal and opposite because 1. The two forces
form an action-reaction pair 2. The net force
on the car is zero 3. Neither of the above
The two forces cannot be an action-reaction pair
because they act on the same object (car). Car is
at rest - therefore, it has no net forces acting
on it. The forces acting on it add up to zero
The Normal Force
When person is holding the bag above the table he
must supply a force. When the bag is placed on
the table, the table supplies the force that
holds the bag on it That force is perpendicular
or normal to the surface of the table
Do You Feel The Normal Force?
Yes, you can feel an upward force on your feet. F
gm 9.8x100 980 Newtons! That force is
spread out over the area of you foot so its not
so bad. Pressure P F/Area N/m2
Action-Reaction Pairs vs forces acting on an
  • is the normal force, the force the table
    exerts on the TV
  • is perpendicular to the surface
  • is the reaction force the TV exerts on the
  • is force the Earth exerts on object
  • is force object exerts on the earth

Forces Acting on an Object
  • Newtons Law uses the forces acting on an object
  • are acting on the object
  • are acting on other objects

Summary of Newtons laws
  • Newtons First Law
  • The motion of an object does not change unless it
    is acted on by a net external force
  • Newtons Second Law
  • Newtons Third Law

Applications of Newtons Laws
  • Assumptions
  • Objects behave as particles
  • can ignore rotational motion (for now)
  • Masses of strings or ropes are negligible
  • Interested only in the forces acting on the
    object or the system of interest
  • can neglect reaction forces

Example 12
Consider a person standing in an elevator that is
accelerating upward. The upward normal force N
exerted by the elevator floor on the person
is a) larger than b) identical to c) less
than the downward weight W of the person.
Person is accelerating upwards - net upwards
force is non zero
Frictional Force
  • Friction
  • Opposes motion between systems in contact
  • Parallel to the contact surface
  • Depends on the force holding the surfaces
  • Normal force (N)
  • Static friction
  • Force required to move a stationary object
  • fs is less than or equal to µs N
  • Object remains stationary
  • Kinetic friction
  • Frictional force on an object in motion
  • Is generally less than static friction
  • Note Equation contains only magnitudes of forces
    since friction and normal force have different

µS coefficient of static friction µK
coefficient of kinetic friction
Friction (II)
  • Static friction acts to keep the object from
  • If F increases, so does ƒs
  • If F decreases, so does ƒs
  • ƒs µs n
  • The force of kinetic friction acts when the
    object is in motion
  • ƒk µk n
  • Variations of the coefficient with speed will be

Example 13
You are pushing a wooden crate across the floor
at constant speed. You decide to turn the crate
on end, reducing by half the surface area in
contact with the floor. In the new orientation,
to push the same crate across the same floor with
the same speed, the force that you apply must be
about a) four times as great b) twice as
great c) equally as great d) half as
great e) one-fourth as great as the force
required before you changed the crate orientation.
Frictional force does not depend on the area of
contact. It depends only on the normal force and
the coefficient of friction for the contact.
Example 14A
  • You are driving a car up a hill with constant
    velocity. On a piece of paper, draw a Free Body
    Diagram (FBD) for the car. How many forces are
    acting on the car? 12345

weight/gravity (W)normal (FN)engine/motor
(Fcar_on_road(action) gt (reaction) Froad on car)
Example 14B
  • The net force on the car is 1. Zero 2. Pointing
    up the hill 3. Pointing down the hill 4.
    Pointing vertically downward 5. Pointing
    vertically upward

SF ma 0
Example 14C
  • You are driving a car up a hill with constant
  • How many forces are acting on the car?

weight/gravity (W)normal (FN)engine/motor
(Fcar_on_road(action) gt (reaction) Froad on car)
Example 14D
  • You are driving a car up a hill with constant
  • The net force on the car is now1. Zero 2.
    Pointing up the hill 3. Pointing down the hill
    4. Pointing vertically downward 5. Pointing
    vertically upward

Froad on car
Example 14- Summary
  • Often important to resolve the weight into
    components parallel and perpendicular to the
  • Then if Fw parallel Froad on car
  • Constant velocity
  • if Fw parallel lt Froad on car
  • Accelerate up the hill
  • If Fw parallel gt Froad on car
  • Accelerate down the hill
  • Fw parallel gt fmax FN µs µsMgcosf
  • Slide down the hill

Tension can be transmitted around corners If
there is no friction in the pulleys, T remains
the same
Tension is a force along the length of a medium
More on tension
For massless cords passing over frictionless
pulleys or surfaces the whole rope is
characterized by a single tension, which is
usually denoted as T. If a rope is in tension,
then at any cross section along its length, the
left part pulls on the right by a force T and the
right side pulls on the left by a force, T.
Hence 1. There is a single tension, T,
characterizing an ''ideal'' cord. 2. A rope can
only pull along its length. It never pushes and
it never exerts a force perpendicular to its
length. Rule 1) sets the magnitude of the forces
produced by a cord and rule 2) determines the
direction of the force produced on an object in
contact with the cord.
Example 14 Pulley I
  • What is the tension in the string?
  • A) TltW
  • B) TW
  • C) WltTlt2W
  • D) T2W

Same answer
Example 15 Pulley II
  • What is the tension in the string?
  • A) TltW
  • B) TW
  • C) WltTlt2W
  • D) T2W

Example 15
In the 17th century, Otto von Guricke, a
physicist in Magdeburg, fitted two hollow bronze
hemispheres together and removed the air from the
resulting sphere with a pump. Two eight-horse
teams could not pull the halves apart even though
the hemispheres fell apart when air was
readmitted! Suppose von Guricke had tied both
teams of horses to one side and bolted the other
side to a heavy tree trunk. In this case, the
tension on the hemisphere would be a) twice
what it was b) exactly what it was c) half
what it was