Motion with constant acceleration

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Motion with constant acceleration
Lecture deals with a very common type of motion
motion with constant acceleration After this
lecture, you should know about Kinematic
equations Free fall.
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Summary of Concepts(from last lecture)

• kinematics A description of motion
• position your coordinates
• displacement ?x change of position
• distance magnitude of displacement
• velocity rate of change of position
• average ?x/?t
• instantaneous slope of x vs. t
• speed magnitude of velocity
• acceleration rate of change of velocity
• average ?v/?t
• instantaneous slope of v vs. t

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Motion with constant acceleration in 1D
Kinematic equations
An object moves with constant acceleration when
the instantaneous acceleration at any point in a
time interval is equal to the value of the
average acceleration over the entire time
interval. Choose t00
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Motion with constant acceleration in 1D
Kinematic equations (II)
Because velocity changes uniformly with time, the
average velocity in the time interval is the
arithmetic average of the initial and
final velocities
(1)
(2)
Putting (1) and (2) together
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Motion with constant acceleration in 1D
Kinematic equations (III)
The area under the graph of velocity vs time for
a given time interval is equal to the
displacement ?x of the object in that time
interval
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Motion with constant acceleration in 1D
Kinematic equations (IV)
Putting the following two formulas together
another way
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Motion with constant acceleration in 1D
Kinematic equations (V)

• ?x v0t 1/2 at2 (parabolic)
• ?v at (linear)
• v2 v02 2a ?x (independent of time)

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Use of Kinematic Equations

• Gives displacement as a function of velocity and
  time
• Use when you dont know or need the acceleration

• Shows velocity as a function of acceleration and
  time
• Use when you dont know or need the displacement

• Gives displacement given time, velocity
  acceleration
• Use when you dont know or need the final velocity

• Gives velocity as a function of acceleration and
  displacement
• Use when you dont know or need the time

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Example for motion with aconst in 1D Free fall
The Guinea and Feather tube
Experimental observations
Earths gravity accelerates objects equally,
regardless of their mass.
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Free Fall Principles

• Objects moving under the influence of gravity
  only are in free fall
• Free fall does not depend on the objects
  original motion
• Objects falling near earths surface due to
  gravity fall with constant acceleration,
  indicated by g
• g 9.80 m/s2
• g is always directed downward
• toward the center of the earth
• Ignoring air resistance and assuming g doesnt
  vary with altitude over short vertical distances,
  free fall is constantly accelerated motion

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Summary Constant Acceleration

• Constant Acceleration
• x x0 v0xt 1/2 at2
• vx v0x at
• vx2 v0x2 2a(x - x0)
• Free Fall (a -g)
• y y0 v0yt - 1/2 gt2
• vy v0y - gt
• vy2 v0y2 - 2g(y - y0)

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Example 1

• A ball is thrown straight up in the air and
  returns to its initial position. For the time
  the ball is in the air, which of the following
  statements is true?
• 1 - Both average acceleration and average
  velocity are zero.
• 2 - Average acceleration is zero but average
  velocity is not zero.
• 3 - Average velocity is zero but average
  acceleration is not zero.
• 4 - Neither average acceleration nor average
  velocity are zero.

Free fall acceleration is constant (-g) Initial
position final position ?x0 averaged vel
?x/ ?t 0
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Free Fall dropping throwing

• Drop
• Initial velocity is zero
• Acceleration is always g -9.80 m/s2
• Throw Down
• Initial velocity is negative
• Acceleration is always g -9.80 m/s2
• Throw Upward
• Initial velocity is positive
• Instantaneous velocity at maximum height is 0
• Acceleration is always g -9.80 m/s2

vo 0 (drop) volt 0 (throw) a g
v 0 a g
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Throwing Down Question
A ball is thrown downward (not dropped) from the
top of a tower. After being released, its
downward acceleration will be 1. greater than
g 2. exactly g 3. smaller than g
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Example 2

• A ball is thrown vertically upward. At the very
  top of its trajectory, which of the following
  statements is true
• 1. velocity is zero and acceleration is zero2.
  velocity is not zero and acceleration is zero3.
  velocity is zero and acceleration is not zero4.
  velocity is not zero and acceleration is not zero

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Example 3A

• Dennis and Carmen are standing on the edge of a
  cliff. Dennis throws a basketball vertically
  upward, and at the same time Carmen throws a
  basketball vertically downward with the same
  initial speed. You are standing below the cliff
  observing this strange behavior. Whose ball is
  moving fastest when it hits the ground?
• 1. Dennis' ball 2. Carmen's ball 3.
  Same

On the dotted line ?y0 gt v2 v02 v
v0 When Denniss ball returns to dotted
line its v -v0 Same as Carmens
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Example 3B

• Dennis and Carmen are standing on the edge of a
  cliff. Dennis throws a basketball vertically
  upward, and at the same time Carmen throws a
  basketball vertically downward with the same
  initial speed. You are standing below the cliff
  observing this strange behavior. Whose ball hits
  the ground at the base of the cliff first?
• 1. Dennis' ball 2. Carmen's ball 3.
  Same

Time for Denniss ball to return to the dotted
line v v0 - g t v -v0 t 2 v0 / g This
is the extra time taken by Denniss ball
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Example 4

• An object is dropped from rest. If it falls a
  distance D in time t then how far will if fall in
  a time 2t ?
• 1. D/4 2. D/2 3. D 4. 2D
  5. 4D

Follow-up question If the object has speed v at
time t then what is the speed at time 2t ? 1.
v/4 2. v/2 3. v 4. 2v
5. 4v
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Example 5
Which of the following statements is most nearly
correct? 1 - A car travels around a circular
track with constant velocity. 2 - A car travels
around a circular track with constant speed. 3-
Both statements are equally correct.

• The direction of the velocity changes when going
  around circle.
• Speed is the magnitude of velocity -- it does not
  have a direction and therefore does not change

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Motion in 2D
After this lecture, you should know
about Vectors. Displacement, velocity and
acceleration in 2D. Projectile motion.
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One Dimension

• Define origin
• Define sense of direction
• Position is a signed number (direction and
  magnitude)
• Displacement, velocity, acceleration are also
  specified by signed numbers

Reference Frame
0
1
2
3
4
-4
-1
-2
-3
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Vectors

• There are quantities in physics which are
  determined uniquely by one number
• Mass is one of them.
• Temperature is one of them.
• Speed is one of them.
• We call those scalars.
• There are others where you need more than one
  number for instance for 1D motion, velocity has
  a certain magnitude--
• that's the speed--
• but you also have to know whether it goes this
  way or that.
• So there has to be a direction.
• We call those vectors.

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Two Dimensions

• Again, select an origin
• Draw two mutually perpendicular lines meeting at
  the origin
• Select /- directions for horizontal (x) and
  vertical (y) axes
• Any position in the plane is given by two signed
  numbers
• A vector points to this position

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Properties of vectors

• Equality of two Vectors
• Two vectors are equal if they have the same
  magnitude and the same direction
• Movement of vectors in a diagram
• Any vector can be moved parallel to itself
  without being affected
• Negative Vectors
• One vector is the negative of another one if they
  have both the same magnitude but are 180 apart
  (opposite directions)
• Resultant Vector
• The resultant vector is the sum of a given set of
  vectors
• Position can be anywhere in the plane

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Adding and subtracting vectors geometrically
y
D
x
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Multiplying or Dividing a Vector by a Scalar

• The result of the multiplication or division is a
  vector
• The magnitude of the vector is multiplied or
  divided by the scalar
• If the scalar is positive, the direction of the
  result is the same as of the original vector
• If the scalar is negative, the direction of the
  result is opposite that of the original vector

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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
• The y-component of a vector is the projection
  along the y-axis
• Then, one can define the component vectors
• Attention ? is measured counter-clock-wise with
  respect to the positive x-axis

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Components of a vector (II)

• The components are the legs of the right triangle
  whose hypotenuse is
• May still have to find ? with respect to the
  positive x-axis

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Adding Vectors Algebraically

• Choose a coordinate system and sketch the vectors
• Find the x- and y-components of all the vectors
• Add all the x-components
• This gives Rx
• Add all the y-components
• This gives Ry
• Use the Pythagorean Theorem to find the magnitude
  of the resultant
• Use the inverse tangent function to find the
  direction of R
• Inversion is not unique, the value will be
  correct only if the angle lies in the first or
  fourth quadrant
• In the second or third quadrant, add 180

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Example 6

• Can a vector have a component bigger than its
  magnitude?
• Yes
• No

The square of magnitude of a vector is given in
terms of its components by R2 Rx 2 Ry 2 Since
the square is always positive the components
cannot be larger than the magnitude
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Example 7

• The sum of the two components of a non-zero 2-D
  vector is zero. Which of these directions is the
  vector pointing in?
• 45o
• 90o
• 135o
• 180o

135o
-45o
The sum of components is zero implies Rx -
Ry The angle, ? tan-1(Ry / Rx) tan-1 -1
135o -45o (not unique, multiples of 2 ?)
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2D motion Displacement

• The position of an object is described by its
  position vector,
• The displacement of the object is defined as the
  change in its position

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2D motion Velocity and acceleration

• The average velocity is the ratio of the
  displacement to the time interval for the
  displacement
• The instantaneous velocity is the limit of the
  average velocity as ?t approaches zero
• The direction of the instantaneous velocity is
  along a line that is tangent to the path of the
  particle and in the direction of motion
• The average acceleration is defined as the rate
  at which the velocity changes
• The instantaneous acceleration is the limit of
  the average acceleration as ?t approaches zero
• Ways an object might accelerate
• The magnitude of the velocity (the speed) can
  change
• The direction of the velocity can change
• Both the magnitude and the direction can change

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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

x and y motions are independent! They share a
common time t
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2D motion Projectile motion
Dimensional Analysis
Motion of a soccer ball
Strategy
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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
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Projectile Motion
y -x2, i.e. parabolic dependence on x
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Projectile MotionMaximum height reachedTime
taken for getting there
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Projectile Motion Maximum Range
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Projectile Motion at Various Initial Angles

• Complementary values of the initial angle result
  in the same range
• The heights will be different
• The maximum range occurs at a projection angle of
  45o

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Soccer Ball
Make sense of what you get
Check limiting cases