OneDimensional Motion in the Vertical Direction y axis or Freely Falling Bodies

1
One-Dimensional Motionin the Vertical Direction
(y axis)orFreely Falling Bodies

• Montwood High School
• Physics
• R. Casao

2
Freely Falling Bodies

• Any object that is released into the air is
  subject to a vertical acceleration due to the
  attractive gravitational force between the object
  and the Earth.
• The direction of the acceleration is downward
  toward the surface of the Earth.
• For now, we will only consider objects that are
  moving either straight up or straight down.

3

• The vertical acceleration due to gravity has the
  value of 9.8 m/s2 near the surface of the Earth.
• Most books and people use the letter g to
  indicate the acceleration due to gravity.
• ag 9.8 m/s2
• The magnitude of ag decreases with increasing
  height above the Earths surface.
• The vector ag is directed downward toward the
  center of the Earth. Down is positive for ag up
  is negative for ag.

4
Air Resistance

• Air resistance is the flow of air around an
  object and it acts to slow the velocity of a
  moving object.
• We will ignore the effect of air resistance on
  the objects.
• In the absence of air resistance, all objects
  dropped near the surface of the Earth change
  velocity with a constant acceleration.
• All objects travel the same distance in the same
  amount of time (which is also true in the
  presence of air resistance as long as the air
  resistance on both objects is the same). All
  objects released from the same height at the same
  time will strike the ground at the same time.
• This motion is called free fall.

5

• A freely falling object is any object moving
  freely under the influence of gravity.
• Objects thrown upward or downward and those
  released from rest (dropped) are all freely
  falling objects once they are released.
• An object thrown upward or downward will
  experience the same acceleration as an object
  released from rest.
• Once they are in free fall, all objects will have
  an acceleration downward of -9.8 m/s2.

6

• When water is inside a container and the
  container has a hole in the side, the water will
  escape out the side of the hole.
• Gravity pulls downward on both the container and
  the water in the container. The hand pulls
  upward on the container to keep it from falling.
  The bottom of the container pushes upward on the
  water directly over it. The sides of the
  container help to keep the water in the container
  against the pull of gravity. At the location of
  the hole, the container cannot keep the water in
  and gravity pulls it out.
• When the container is dropped, both the container
  and the water in it are in free fall and are
  accelerating downward. The pull of gravity is
  the same on both the container and the water and
  the water does not come out of the container.

7

• When air is in the tube, air resistance acts on
  both the rock and the paper. Air resistance has
  a greater effect on the paper, therefore, the
  rock strikes the bottom first.
• When air is removed from the tube, there is no
  air resistance to act on either the rock or the
  paper and both the rock and the paper strike the
  bottom at the same time.

8

• Free-falling objects are in a state of
  acceleration.
• They are accelerating at a rate of 9.8 m/s2. This
  means that the velocity of a free-falling object
  is changing by 9.8 m/s every second.
• If dropped from a position of rest, the object
  will be traveling 9.8 m/s at the end of the first
  second, 19.6 m/s at the end of the second second,
  29.4 m/s at the end of the third second, etc.

9

• This is a flash photograph of two balls released
  at the same time. Each image of the balls
  represents an equal time period of 1/30th of a
  second.   Both balls are falling freely, but one
  ball (the ball on the right) was projected
  horizontally.  
• Notice that the horizontal displacement of this
  ball does not change in equal periods of time.  
  The ball moves an equal distance to the right
  for each successive time period.

10

• In the horizontal direction there is no external
  force and therefore no acceleration. We can see
  that the horizontal velocity of the projected
  ball is constant.
• In the vertical direction, there is a noticeable
  acceleration.   We can see this because for each
  successive time period, the vertical displacement
  of each ball increases.   The change in vertical
  displacement is the same for both.   Therefore,
  the vertical velocities for each will be equal.

11

• For an object which is released from one point,
  rises upward to a maximum height, then falls
  downward to return to its initial position
• Time up time down
• Total time time up time down
• vyi -vyf
• The velocity at the highest point is 0 m/s.
• The acceleration at every point, including the
  highest point, is 9.8 m/s2.

12

• For an object thrown into the air and under the
  acceleration of gravity, the velocity at a point
  on the way up and the velocity at the same point
  on the way down are equal in magnitude, but
  opposite in direction.
• Up v 30 m/s
• Down v - 30 m/s

13
Equations

• If we neglect air resistance and assume that the
  gravitational acceleration does not change much
  with altitude, the motion of a freely falling
  body can be described as motion in one-dimension
  with constant acceleration.
• The vertical direction will be the y-axis with up
  being positive and down being negative.
• Using the equations for acceleration along the
  x-axis and replacing x with y and a with ag

14
The negative signs in these equations indicate
the direction of the gravitational acceleration.
15

• If the direction of motion is upward, then ?y is
  positive if the direction of motion is downward,
  then ?y is negative.
• For an object thrown upward
• Final velocity is 0 m/s (at the highest point).
• Acceleration ag 9.8 m/s2 downward.
• Initial velocity is greater than 0 m/s.
• Maximum height will be equal to ?y.
• For an object that is falling downward
• Initial velocity is 0 m/s if the object is
  dropped.
• Initial velocity is greater than 0 m/s and is
  negative if the object is thrown downward.

16

• Acceleration ag 9.8 m/s2 downward.
• Distance object falls is negative and is equal to
  ?y.
• The final velocity when the object first contacts
  the ground will be negative and will be greater
  than 0 m/s. Vyf is the velocity the object has
  as it first makes contact with the surface and it
  is NOT 0 m/s. The object will eventually slow
  down from vyf to 0 m/s as it interacts with the
  surface, but the object first strikes the surface
  with velocity vyf.
• Velocities in the upward direction are positive.
• Velocities in the downward direction are
  negative.

17

• For an object that is falling downward, the ?y is
  negative. When using the vyf2
  vyi2 (2ag?y) equation, the final velocity
  will be positive because you cannot take the
  square root of a negative number. You will have
  to add the negative sign to show that the object
  is moving downward.
• For an object that is thrown upward from a point
  and then lands at that point
• vyi - vyf time up time down
• Total time time up time down
• Distance up distance down

18

• All of this is true because the object slows down
  (deceleration) on the way up at the same rate as
  it speeds up on the way down. Both the
  deceleration and the acceleration are constant
  and due to the force of gravity pulling on the
  object.
• For Problems Involving Ascending (Rising) or
  Descending (Moving Downward) Objects
• These types of problems have an item being
  released or falling off of an object (I will use
  a hot air balloon as an example) that is either
  rising or coming down with some speed. The key
  to solving these problems is to realize that the
  balloon and any object in the balloon or attached
  to the balloon is traveling at the same speed at
  the balloon.

19

• For objects that are rising
• If a balloon is rising at 10 m/s and an object is
  released from the balloon. The released object
  will have an upward velocity of 10 m/s and will
  be decelerated by gravity as it travels upward.
  The final velocity on the way up is 0 m/s. The
  upward distance ?y traveled by the object will be
  positive.
• Both the upward distance ?y and the time up can
  be determined from this information.

20

• After the object reaches its highest point,
  gravity will begin to accelerate the object down
  toward the ground. The initial velocity is 0 m/s
  and the distance the object falls will be the
  distance above the ground at which the object
  left the balloon plus the upward distance ?y
  traveled by the object to the highest point. The
  ?y down will be negative to indicate that the
  motion is downward.

21

• From this information, you can determine the time
  down and the velocity at which the object will
  strike the surface (vyf). The final velocity
  should be negative to indicate that the direction
  of motion is downward.
• The total time in the air will be equal to the
  time up the time down.
• The maximum height will be equal to the height of
  release ?y up. 

22

• For objects that are hovering (staying on one
  place not rising or moving downward)
• The initial velocity will be zero and you can
  solve this problem as you would for an object
  that is dropped downward. The ?y down will be
  negative to indicate that the motion is downward.
  
• From this information, you can determine the time
  down and the velocity at which the object will
  strike the surface (vf). The final velocity
  should be negative to indicate that the direction
  of motion is downward.

23

• For objects that are descending (moving
  downwards) when the object is released
• If a balloon is descending at 10 m/s and an
  object is released from the balloon. The
  released object will have a downward velocity of
  -10 m/s and will be accelerated by gravity as it
  travels downward. The initial velocity vi and
  the ?y down will be negative to indicate that the
  motion is downward.
• From this information, you can determine the time
  down and the velocity at which the object will
  strike the surface (vyf). The final velocity
  should be negative to indicate that the direction
  of motion is downward.