The 5 Kinematic Equations (Only for constant acceleration!)

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The 5 Kinematic Equations (Only for constant
acceleration!)
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(1)
(2)
(3)
(4)
(5)
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Problem Solution Guidelines
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• Draw a sketch
• Indicate origin and positive direction
• List the given and unknown quantities using the
  symbols of the equations. (xo, x, vo, v, a, t)
• Is time known or do we need to find it?
• What are we to solve for?
• Look at the kinematic equations and see if there
  is one where we know everything except the one
  quantity that we are looking for.

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More Guidelines
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• Rewrite the equations using the known quantities.
• Look at the knowns and unknowns and map a
  strategy of solution.
• Check your units
• Make sure you are answering the question.

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A cars position at t 0 s is -100 m from the
origin. It starts from rest at this time
(meaning v0 0 m/s) and uniformly accelerates at
a rate of 5 m/s2 in the x direction. Answer the
following questions.
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(A) Calculate the cars velocity at t 8 s.
Assume that the car accelerates at a constant
rate the entire time. (B) Calculate the cars
position at t 8 s. Make the same assumption as
in (A). (C) Calculate the cars displacement
during this time interval. (D) Calculate the
cars average velocity during this time
interval. (E) Write equations for the cars
instantaneous position and velocity as a
function of time t. That is, write equations
for x(t) and v(t). (F) Make a table of the cars
position and velocity for times t 0, t 1 s, t
2 s, t 8 s. Then make a graph of x-t and
v-t for the car.
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Key information A cars position at t 0 s is
-100 m from the origin. It starts from rest at
this time (meaning v0 0 m/s) and uniformly
accelerates at a rate of 5 m/s2 in the x
direction.
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(A) Calculate the cars velocity at t 8 s.
Assume that the car accelerates at a constant
rate the entire time.
We were given that v(0) 0, a 5 m/s2 and t 8
s.
v(t) v(0) a t
v(8) 0 (5 m/s2)(8 s)
ANSWER v(8) 40 m/s x
The cars speed is 40 m/s in the x direction.
It accelerated, or gained speed (because both v
and a were ) at a rate of 5 m/s per second for 8
seconds.
(B) Calculate the cars position at t 8 s.
We also were given that x(0) -100 m.
x(t) x(0) v(0) t (1/2) a t2
x(8) -100 m (0)(8 s) (1/2) (5 m/s2) (8 s)2
The car moved from -100 m to 60 m during those 8
seconds.
ANSWER x(8) 60 m
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Key information A cars position at t 0 s is
-100 m from the origin. It starts from rest at
this time (meaning v0 0 m/s) and uniformly
accelerates at a rate of 5 m/s2 in the x
direction.
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(C) Calculate the cars displacement during this
time interval.
We nearly answered this question in part (B).
?x x(8) - x(0)
?x 60 m - (-100 m)
ANSWER ?x 160 m
CAUTION Dont forget your signs here.
(D) Calculate the cars average velocity during
this time interval.
Substituting the given values into the equation
yields...
v 160 m / 8 s
ANSWER v 20 m/s x
We could have also found the average velocity by
using the formula.
v ( v(0) v(8) ) / 2
v ( 0 40 m/s ) / 2 20 m/s
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Key information A cars position at t 0 s is
-100 m from the origin. It starts from rest at
this time (meaning v0 0 m/s) and uniformly
accelerates at a rate of 5 m/s2 in the x
direction.
(E) Write equations for the cars instantaneous
position and velocity as a function of time
t. That is, write equations for x(t) and v(t).
Recall that we were given that x(0) -100 m,
v(0) 0, a 5 m/s2
v(t) v(0) a t substituting yields
ANSWER v(t) 5 t
x(t) x(0) v(0) t (1/2) a t2 substituting
yields
x(t) -100 (0)(t) (½)(5) t2
ANSWER x(t) -100 2.5 t2
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Key information A cars position at t 0 s is
-100 m from the origin. It starts from rest at
this time (meaning v0 0 m/s) and uniformly
accelerates at a rate of 5 m/s2 in the x
direction.
(F) Make a table of the cars position and
velocity for times t 0, t 1 s, t 2 s, t
8 s. Then make a graph of x-t and v-t for the
car.
Graphs of x-t and v-t on the next slide
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Key information A cars position at t 0 s is
-100 m from the origin. It starts from rest at
this time (meaning v0 0 m/s) and uniformly
accelerates at a rate of 5 m/s2 in the x
direction.
The slope of the v-t line is 5 m/s2.
The acceleration of the car is 5 m/s2. Since v
and a are both , the car gains speed.
The slope of the x-t curve increases. The speed
of the car increases each second. Since v and a
are both , the car gains speed.
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A truck having an initial velocity of 30 m/s
accelerates uniformly at 3 m/s2.
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• What is its velocity after 4 seconds?
• 42 m/s
• How far has it traveled in 4 seconds?
• 144 m
• c) How fast is it moving after it has traveled
  100 m from its original position?
• 38.73 m/s
• How long did it take to travel the 100 m?
• 2.91 s

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A truck covers 40 meters in 8.5 seconds while
smoothly slowing down to a final speed of 2.8 m/s.
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• Find its initial speed.
• 6.61 m/s
• Find its acceleration.
• -0.45 m/s2

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A speedboat moving at 30 m/s approaches a no-wake
buoy marker 100 meters ahead. The pilot slows
the boat with a constant acceleration of 3.5
m/s2 by reducing the throttle.
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• How long does it take the boat to reach the buoy?
• 4.53 s
• b) What is the velocity of the boat when it
  reaches the buoy?
• 14.14 m/s

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Free Fall
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• An object is in free fall when the only force
  acting on it is the force of gravity
• The force of gravity points downward
• Air resistance ignored

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Objects in Free Fall
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• To a physicist, the term "free fall" has a
  different meaning than it does to a skydiver. In
  physics, free fall is the (one-dimensional)
  motion of any object under the influence of
  gravity only - no air resistance or friction
  effects of any kind, whereas it is air resistance
  that makes skydiving a hobby rather than a
  suicide attempt!

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Objects in Free Fall
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• You might think that since just about everything
  we observe falling is falling through the air,
  that "physics free fall" must be a pretty useless
  idea in practice. Not so! Any falling object's
  motion is at least approximately free fall as
  long as

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Objects in Free Fall
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• ... it is relatively heavy compared to its
  size. (Dropping a ball, or jumping off a chair,
  is a free-fall motion, but dropping an unfolded
  piece of paper, or the motion of a dust particle
  floating in the air, is not. If you crumple the
  paper into a "paper wad", however, its motion is
  approximately free fall.
• ... it falls for a relatively short time. (If
  you jump off a chair, you are in free fall. After
  you have jumped out of an airplane and fallen for
  several seconds, you are not in free fall, since
  air resistance is now a factor in your motion.)
• ... it is moving relatively slowly. (If you
  drop a ball or throw it down its motion will be
  free fall. If you shoot it out of a cannon, its
  motion won't be free fall.)

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Objects in Free Fall
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• You should also note that an object doesn't have
  to be falling to be in free fall - if you throw a
  ball upward its motion is still considered to be
  free fall, since it is moving under the influence
  of gravity. 

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Free Fall
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• An object is in free fall when the only force
  acting on it is the force of gravity
• The force of gravity points downward
• Air resistance ignored
• Acceleration due to the force of gravity near
  the surface of Earth is downward and has a value
  of g 9.8 m/s2
• (note g is just the magnitude of the objects
  acceleration)
• We have then the conditions of one-dimensional
  kinematics straight line motion with constant
  acceleration.

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Sample Problem
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• A ball is thrown vertically upward at 10 m/s.
  How high will it get, how long will it be in the
  air, and how fast will it be moving when it hits
  the ground.