Motion With Uniform Acceleration

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Motion With Uniform Acceleration

• Week 3 -- Lesson 1

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Objectives
By the end of this lesson, you should be able to

• Recognize the expressions that relate
  displacement, velocity, and acceleration for
  motion that involves an object undergoing a
  constant acceleration. Also, be able to state
  the conditions under which those expressions are
  applicable.
• Realize that in situations involving motion along
  the vertical direction there is a natural
  downward acceleration due to gravity and be able
  to use this fact when solving problems.

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Assumptions In Our Straight-Line Kinematics Model

• In our model to describe the motion of an object
  we will make the following assumptions
• Object has a constant acceleration
• Object is moving in a straight-line
• The only type of motion we will ever be able to
  model is straight-line motion. More complicated
  motions will be made up of combinations of
  straight-line motions

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A Model For Uniformly Accelerated Motion

• We begin our discussion by trying to determine
  relationships between the parameters associated
  with the motion of an object displacement,
  velocity, acceleration, and time. We start, by
  trying to determine a relationship between the
  velocity of an object and the time of travel for
  the object.

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Relationship Between Velocity and Time
We assume that
, then
Object has a velocity, v, at time t
Object has an initial velocity, vo, when t 0 s
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Relationship BetweenVelocity and Time
Since a is constant, it can be pulled out of the
integral
Or, in words, the velocity of an object at the
end of a time interval is equal to the velocity
at the beginning of the interval plus the
velocity change of the object during the time
interval.
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Displacement and Time
To get a relationship between the displacement
of an object and the time of travel
Object is at position x at time t
Object is at xxo at t 0s
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Displacement and Time
(vo and a are constants.)
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Velocity and Displacement
To this point, we have discussed relationships
where time was one of the parameters
involved. In some cases, you might want to relate
the velocity of an object to its displacement (an
example where this might be of interest is if you
were drag racing). How do we relate velocity and
displacement without involving time when all we
have discussed are time derivatives? We must
make a change of variables in our relationship
between velocity and acceleration
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Velocity and Displacement
Multiply by 1
Multiplication is commutative.
Since v dx/dt
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Velocity and Displacement
object is moving with velocity v when located at x
object is moving at vo when located at xo
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Velocity and Displacement
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Summary
Relationship Relates
velocity and time
position and time
velocity and position
Restrictions
Object is moving in a straight line with a
constant acceleration.
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Summary Continued

• Notice that the right hand column on the previous
  slide does not list the initial position, initial
  velocity, or acceleration. This is because each
  of these three parameters are constants in this
  model. So, the way you select from the three
  expressions on the previous slide is to analyze
  the information you are given and the information
  you are asked for. If you were given position
  information and asked for velocity information,
  then you would choose the third expression.

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Summary Continued

• Signs are extremely important when using the
  straight-line kinematics formulas. Make sure that
  you pay attention and carry the sign for the
  correct direction of each of the vector
  quantities.
• If you run a stop sign while driving, a police
  officer will pull you over and ask Did you see
  that sign back there? It is pretty important.
  Mathematical signs are just as important when it
  comes to successful problem solving.

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Example

• An object is moving to the left with a speed of
  5.0 m/s when it undergoes a constant acceleration
  of 1.5 m/s2, also to the left. How long does it
  take the car to reach a velocity of 9.3 m/s to
  the left?

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Solution

• Work solution in class

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Speeding Up And Slowing Down

• Notice that the acceleration that was plugged
  into the relationship between velocity had a
  negative sign. This sign was the result of the
  acceleration being directed along the negative x
  axis. Many students will incorrectly associate a
  negative acceleration with an object slowing
  down. Notice in the last example that the object
  actually sped up although the acceleration was
  negative. So, how can you tell if the speed of an
  object will increase or decrease?

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Speeding Up Or Slowing Down

• To determine if the speed of an object will
  increase or decrease, look at the relative
  directions of the instantaneous velocity and
  acceleration vectors.
• To speed up, the acceleration and instantaneous
  velocity vectors must be in the same direction.
• To slow down, the acceleration and instantaneous
  velocity vectors must be in the opposite direction

v
a
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Speeding Up Or Slowing Down

• You can think of the acceleration vector as the
  velocity change vector (after all, acceleration
  is the time rate of change of velocity). Thus,
  if the velocity change is in the same direction
  as the original velocity vector, that vector will
  get longer, or the magnitude will increase.
  Conversely, if the velocity change is in the
  opposite direction as the original velocity
  vector, that vector will get shorter, or the
  magnitude will decrease.

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Reminder

• Always pay attention to what you are asked for
  when solving problems. When reporting an answer,
  check to see if the quantity you are asked for is
  a scalar or vector quantity. You want to make
  sure that you put a direction on vector
  quantities. Conversely, you should not put a
  direction on a scalar quantity.

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Try It On Your Own

• A car has a maximum braking acceleration of 5.2
  m/s2. What is the maximum speed that this car
  can be traveling with if it is to stop before an
  intersection that is 40.0 m away when the driver
  notices the traffic light change to red?

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More Complex Situations

• To this point, we have discussed fairly simple
  situations motions where the object moved with a
  single acceleration for the entire path and where
  enough information was known to use a single
  expression. Many times in the real world,
  situations are not this simple. How do you handle
  these more complicated situations? Think of it
  like trying to eat a 16 pizza. You dont eat
  the pizza in a single bite. Rather, you break
  (cut) it up into pieces that you can handle.

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More Complex Situations

• How do you decide to break up the motion?
  Remember that the expressions we developed were
  designed for motion where the acceleration of the
  object is constant. So, break the motion into
  intervals where the acceleration of the object is
  constant over the entirety of each interval.
• For single interval motions where not enough
  information is given to use a single expression
  to solve for a certain quantity, use one of the
  expressions to solve for a different parameter
  and then one of the other two expressions to
  solve for the desired quantity, as in the next
  example.

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Example

• A car accelerates uniformly from 10 m/s to 25 m/s
  in a time of 4.25 s. What distance did the car
  travel in the 4.25 s interval?

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Solution To Example

• Solution to last example.

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Try It On Your Own

• An object starts from rest and accelerates
  uniformly to the right to a speed of 5.0 m/s in
  3.2 s, then travels at constant speed for 12.0 s,
  and finally decelerates to rest in a time of 5.0
  s. What is the displacement of the object from
  the starting point at the end of the trip?

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Freely Falling Bodies

• There is a natural acceleration in the vertical
  direction
• ay 9.8 m/s2 downward
• Problems involving vertical motion are solved
  exactly the same way as problems involving
  horizontal motion

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Freely Falling Bodies Continued

• The text rewrites the equations for motion with a
  constant acceleration for the case of freely
  falling bodies. Since there is no difference in
  the way that one should attack a problem
  involving motion with constant acceleration along
  the horizontal axis and along the vertical axis,
  I suggest working them the same way with the same
  expressions. On the next slide I have written our
  expressions with axis subscripts. Then to write
  the expressions for vertical motion, simply
  change all the xs to ys.

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Motion With Constant Acceleration
Horizontal Motion
Vertical Motion
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Example

• A ball is thrown upward at 7.5 m/s. How high
  does it travel? What is its velocity 1.25
  seconds after it is thrown? How long does it take
  the ball to return to the point from which it was
  thrown?

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Solution

• Motion Diagrams
• Knowns and Unknowns
• Sign checks
• Constants
• Motion Graphs

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Summary

• The model we have developed only describes the
  motion of objects moving in a straight line with
  a constant acceleration.
• The only difference between motion along a
  horizontal axis and motion along a vertical axis
  is that there is a natural acceleration along the
  vertical axis.
• When working complex problems, break the problem
  up into parts (or intervals) you can handle

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Assignments

• Chapter 2 pg 23 Freely Falling Bodies
• Pg. 29 Prb. 40-65 odd
• Handout Videopoint Lab on Falling Bodies