Falling Objects and Projectile Motion

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Falling Objects and Projectile Motion

• Chapter 3

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Acceleration Due to Gravity

• What happens if we drop an object?
• Does the object accelerate or fall with a
  constant speed?
• Do two objects behave differently if they have
• different masses?
• different shapes?

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Measuring Gravity Acceleration

• Galileo was the first one to accurately measure
  the acceleration due to gravity.
• To measure this acceleration at that time, the
  action needs to be slowed down.
• By rolling objects down an inclined plane,
  Galileo was able to establish that the
  gravitational acceleration is uniform, or
  constant with time.

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Measuring Gravity Acceleration
Time Position Average velocity
0 s 0 cm
0.05 s 1.2 cm 24 cm/s
0.10 s 4.8 cm 72 cm/s
0.15 s 11.0 cm 124 cm/s
0.20 s 19.7 cm 174 cm/s
0.25 s 30.6 cm 218 cm/s
0.30 s 44.0 cm 268 cm/s
0.35 s 60.0 cm 320 cm/s
Calculate the acceleration due to the gravity
from the graph (g)
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The diagram shows the positions at 0.10-sec
intervals of a ball moving left to right. Is the
ballaccelerated?
Example (Q2p54)
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The diagram shows the positions at 0.05-sec
intervals of two balls moving left to right. Are
either or both of these balls accelerated?
Example (Q3 p54)
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Gravitational Acceleration (g)
Aristotles and Galileos different ideas The
gravitational acceleration for objects near the
surface of the Earth is a constant and has the
value of 9.8 m/s2. It has the same value for
objects with different weight and shape.
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Tracking a Falling Object

• Starting from rest, how does the velocity vary
  with time? (t0.5 s, 1s, )
• How far does the ball fall in different times?
• Assuming air resistance is negligible. The ball
  accelerates at 9.8 m/s2 10 m/s2.

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Tracking a Falling Object

• 10 m/s gt 20 MPH
• 30 m/s 70 MPH

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Throwing a ball downward

• Ball is dropped
• Ball is thrown down

Sample Exercise A ball is throwing down with an
initial velocity of 20m/s. Find (a) the velocity
and (b) the distance traveled at 1-s time
intervals for the first 2s of motion
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Throwing a Ball Upward

• What if the ball is thrown upward?

• The importance is that now the acceleration and
  the initial velocity is in the opposite
  direction.

A ball is thrown upward with an initial velocity
of 20m/s, find its velocity and height in the
first four Seconds. How high can the ball go?
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Throwing a Ball Upward
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Examples Q8 (p54)

• A rock is dropped from the top of a diving
  platform into the swimming pool below. Will the
  distance traveled by the rock in the 0.1 second
  interval near the top of its flight be the same
  as the distance covered in the 0.1 second
  interval just before it hits the water? Explain.

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Examples Q9(p54)

• The graph shows the velocity plotted against
  time for a certain falling object. Is the
  acceleration of this object constant?

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Examples Q10 (p54)

• A ball is thrown downward with a large starting
  velocity.
• Will this ball reach the ground sooner than one
  that is just dropped at the same time from the
  same height?
• Will the ball accelerate more rapidly than one
  that is dropped with no initial velocity?

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Examples

• Q11 (p54) A ball thrown upward moves initially
  with a decreasing velocity.
• What are the directions of the velocity and the
  acceleration vectors during this part of motion?
• Does the acceleration decrease also?

Q 14. A ball is thrown straight upward. At the
very top of its flight, the velocity of the ball
is zero. Is its acceleration also zero?
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Examples E7-9 (p55-56)

• A ball is thrown upward with an initial velocity
  of 15m/s. Using the approximate value of g10
  m/s/s, what are the magnitude and the direction
  of the balls velocity 1 s and 2 s after it is
  thrown?

• How high above the throwing point is the ball in
  above question 1 s and 2 s after it is thrown?

• At what time the ball will reach its highest
  point and what is that height?

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Projectile Motion

• Examples
• Ball running off the table
• Firing a cannon
• Shooting a basket ball
• - an object with an initial velocity moving in
  the air with only the gravitational force acting
  on it
• What does the trajectory look like? (A
  trajectory of a moving object is its actual
  moving path).

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Projectile Motion
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Which of these balls would hit the floor first if
all three left the tabletop at the same time?
Which will hit the floor with the biggest
distance from the table?
Projectile Motion

• Trajectories of a ball with three different
  initial velocities rolling off a table.

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Projectile Motion

• Principles to treat project motion
• (1) First separate the motion in vertical and
  horizontal directions and threat them
  respectively
• Horizontally it is a motion at a constant speed
• Vertically it accelerates downward under the
  influence of gravity.
• (2) The real trajectory of the object is the
  combination of the vertical and the horizontal
  motion.

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Example (Box 3.4 p 48)

• A ball rolls off a tabletop with an initial
  velocity of 3 m/s. If the table is 1.25 m above
  the floor,
• How long does it take for the ball to hit the
  floor?
• How far does the ball travel horizontally?
• Q What determines the time of flight?

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Example (p56

•   E 12. A ball rolls off of a shelf with a
  horizontal velocity of 6 m/s. At what horizontal
  distance from the edge of the shelf does the ball
  land if it takes 0.5 s to reach the floor?
• E14. A ball rolls off of a table with a
  horizontal velocity of 5 m/s. If it takes 0.6 s
  for it to reach the floor
• a. What is the vertical component of the balls
  velocity just before it hits the floor?
• b. What is the horizontal component of the
  balls velocity just before it hits the floor?

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Hitting a Target (Shooting a rifle)

• If the rifle is fired directly at the target in a
  horizontal direction, will the bullet hit the
  center of the target?
• Does the bullet fall during its flight?

• Example (E11, P56). A bullet is fired
  horizontally with an initial velocity of 900m/s
  at a target located 150 m from the rifle .
• How much time is required for the bullet to reach
  the target?
• Using the approximate value of g10 m/s/s. How
  far does the bullet fall in this time? 

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Hitting a Target (throwing a ball and firing a
cannon)

• The trajectory depends on the initial velocity
  and the launch angle.
• The greatest distance is achieved using an angle
  close to 45 if the effects of air resistance are
  negligible.

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Which free-throw trajectory has the greatest
chance of success?
Hitting a Target (Shooting a basket ball)
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Which of the two trajectories shown will result
in a longer time for the ball to reach home
plate?
Example Q23 (p55)
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Example (p56

•   E 16. A projectile is fired at an angle such
  that the vertical component of its velocity and
  the horizontal component of its velocity are both
  equal to 30m/s.
• Using the approximate value of g10m/s/s, how
  long does it take for the ball to reach its
  highest point?
• What horizontal distance does the ball travel in
  this time?