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BY |
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Joy Arakaki |
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Kalpa Bhattarcharjee |
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Rekha Thangellapalli |
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Debbie Lai |
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Pool evolved from a lawn game similar to croquet
played sometime during the 15th century in Northern Europe and in France |
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It was later moved indoors to a wooden table
with green cloth to simulate grass, and a simple border was placed around
the edges. |
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The balls were shoved, rather than struck, with
wooden sticks called "maces" |
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The term "billiard" is derived from
French, from the word "bille", a ball. |
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Velocity |
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Acceleration |
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Force |
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Momentum |
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Impulse |
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Trajectory |
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Work |
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Pool table is flat |
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Initial velocity of all stationary balls is zero |
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Closed system |
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Ignoring air resistance |
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Ignoring friction |
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Velocity is the rate at which an object travels
in a specific direction, thus it is a vector. |
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v=Δs/Δt |
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The absolute value of velocity is speed |
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In Pool… |
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Velocity=(the initial position of the ball - the
finishing position of the ball)/time it took to get there. Also noted is
the direction the ball traveled. |
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A high velocity means the ball covered a great
distance in little time. |
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A high velocity shot is used to break. |
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A low velocity shot is used to nudge a ball |
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Acceleration is the rate at which velocity changes.
Acceleration is a vector quantity and therefore is described as having a
value and a direction |
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a(average acceleration)= Δv/Δt |
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The SI unit used for acceleration is m/s² |
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In pool: |
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Acceleration=(the initial velocity of the ball -
the final velocity of the ball)/time interval. Also noted is the direction
the ball accelerated. |
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The higher the acceleration of the cue ball, the
greater the force it will be able to exert on the other balls. |
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A negative acceleration doesn’t necessarily mean
that the ball is slowing down. It can instead be speeding up in the
opposite direction |
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Force is defined as a push or pull |
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Its unit is the Newton, N |
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1N=1 kg·m/s2 |
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Force directly affects an object’s acceleration |
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F=ma |
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In pool: |
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A player must exert force on the cue stick,
which pushes the cue ball |
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The more force he/she applies, the more the
object will accelerate |
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It is up to the player to judge how much force
to apply |
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In collisions, objects exert equal and opposite
force on each other, |
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which obeys Newton’s third law |
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The momentum of a body is the product of its
mass and velocity |
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Momentum=p=mv |
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The unit for p is kg·m/s |
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Momentum can be transferred from body to body,
which is a transfer of energy |
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The Law of Conservation of Momentum states that
in a closed, isolated system, the total momentum remains constant |
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p+ pb= pa´+ pb´ |
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In pool: |
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The cue ball with momentum collides with an
object ball and transfers some or all of its momentum to the ball |
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Momentum transfers in pool are quite necessary
for desired shots and balls to move |
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When discounting friction, the law of
conservation holds between two balls colliding, the cue stick hitting the
cue ball, and other collisions |
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Initial momentum and final momentum of the
system can be calculated using mass and velocity measurements |
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Here is a closed, isolated system in which
object ball X is moving towards ball Y with an initial momentum, which it
received from the cue ball. Object
ball Y is stationary and it stands directly in front of ball X. |
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Ball X collides with Ball Y and transfers some
of its momentum to it. The total
initial and final momentum in the system is equal. Data was obtained with precise uses of
measuring tape and stopwatch. |
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Before collision: |
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Initial momentum of ball X= px = 0.156·vx |
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Initial momentum of ball Y= py = 0.156·0 = 0 |
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After collision: |
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Final momentum of ball X= px´ = 0.156·vx´ |
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Final momentum of ball Y= py´ = 0.156·vy´ |
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After collecting data, it is evident that the
Law of Conservation of Momentum holds: |
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0.156*vx + 0 = 0.156*vx´ + 0.156*vy´ |
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0.156*1.1+ 0 = 0.156*0.1 + 0.156*1 |
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0.172= 0.172 |
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Collisions can also occur where the balls go off
in different angles. In that case,
the x and y components of the velocity must be multiplied and added
separately. |
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Impulse is defined as the product of the net
force and the time interval over which it acts. It is a vector in the
direction of the force and is measured in units of Newton.second(N.s) |
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Impulse(FΔt)=mΔv |
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However, if the mass of an object remains
constant while there is a change in velocity, then there is a change in
momentum. |
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Therefore, FΔt=Δp (change in momentum) |
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This equation is known as the impulse-momentum
theorem. |
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A large impulse can result from either a large
force acting over a short time or a smaller force acting over a longer
time. |
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However, a large change in momentum only occurs
when there is a large change in impulse |
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In pool: |
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The impulse between the cue ball and a pool ball
determines the change of momentum transferred from the cue ball to the pool
ball. |
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Multiple impulses can be planned between pool
balls in order to transfer momentum to more than one ball |
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The cue stick collides with the initially
stationary cue ball and transfers much momentum. |
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Impulse is involved in this transfer of
momentum. |
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Cue stick exerts force F on the cue ball and cue
ball exerts force -F on the cue stick |
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This occurs for 0.1 seconds |
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The new momentum of the cue ball is equal to the
impulse it received. |
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pcue= F· 0.1 kg·m/s |
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Trajectory is the curved, parabolic path of a
projectile. |
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Equations to describe position: |
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y(t)=yo + (vo)yt + ½ ayt2 |
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x(t)=xo + (vo)xt |
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In billiards… |
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The object ball becomes a projectile in the
“jump shot”. |
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The height of the parabola must be at least the
diameter of the ball being jumped. |
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The width of the parabola must be at least the
distance between the cue ball and object ball plus the distance between the
object ball and the desired landing point. |
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Knowing this, velocity of the cue ball can be
determined. |
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The blue ball must “jump” over the red ball and
hit the black ball. What velocity should the blue ball have? |
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Useful equations: |
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y(t)=yo + (vo)yt + ½ ayt2 |
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x(t)=xo + (vo)xt |
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vy(t)= (vo)y + ayt |
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To find the angle: |
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tan x= 0.0635/0.1778 |
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x>19.65°, so let x=20° |
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There are many combinations of angles, vertical
and horizontal components of velocity to choose. Let’s assume that (vo)x =1
m/s |
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x(t)=xo + (vo)xt |
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0.1778=0 + 1(t) |
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t=.1778 sec. |
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y(t)=yo + (vo)yt + ½ ayt2 |
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0.0635=0+ (vo)y(.1778) + ½(-9.8)(0.1778) 2 |
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(vo)y= 1.228 m/s |
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Initial velocity of blue ball = √(1.2282 +
1) |
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=1.58 m/s [forward, 20° above the
horizontal] |
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Work is the force that is exerted on an object
multiplied by the distance it travels in the direction of the force. |
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W = Fd (unit is Joules) |
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In pool: |
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The direction must be matched in direction
to the right amount of force for an effective shot |
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Even if force is applied in the desired
direction, but the ball goes in a different direction, less work done and
no shot made! |
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Longer shots require more work because a
longer distance is covered |
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Backspin is when the ball rotates opposite the
direction it’s traveling in. |
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Backspin is effective is slowing down the ball’s
speed and reducing the distance needed to stop. |
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It creates greater friction by spinning against
the direction of travel. |
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Backspin can be started by striking the ball at
any point below it’s centerline. |
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If struck above or at the centerline, backspin
can be created by aiming the cue stick downwards. |
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The greater the angle made by the centerline and
cue stick, the higher the spin rate |
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Notes