Projectile Motion

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Projectile Motion

• Describe the motion of an object in TWO
  dimensions
• Keep it simple by considering motion close to the
  surface of the earth for the time being
• Neglect air resistance to make it simpler

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Projectile Motion
The ball is in free fall vertically and moves at
constant speed horizontally!!!
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Projectile Motion
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Projectile Motion
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Projectile Motion
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Projectile Motion
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Projectile Motion
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Projectile Motion
y
x
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Projectile Motion

• What happens when we add air resistance?
• Adds a new force on the ball
• The force is in the opposite direction to the
  balls velocity vector and is proportional to the
  velocity at relatively low speeds
• Need calculus to sort out the resulting motion
• Lowers the angle for maximum range

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Projectile Motion

• The trick to maximum range is just to keep the
  object off the ground for as long as possible.
• This allows the horizontal motion to be a maximum
  since x vxt
• Make range longer by going higher for your
  starting point
• Make range longer by having more velocity

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Projectile Motion
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Satellites

• Curvature of the earth enters into our
  calculations

If I start 5 m above the surface, it will still
be at 5 m after one second if it is moving 8000
m/sec
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Satellites
Throw at 8000 m/sec This is about 18,000
mph Earth circumference is 25,000 miles Takes
25000/18000 1.4 hours 84 minutes Higher
altitude longer
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Satellites
Force of gravity on bowling ball is at 90o to
velocity, so it doesnt change the velocity!!! If
no air resistance, gravity doesnt change speed
of satellite, only direction!!!
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Communications Satellites

• Farther out you go, the bigger the circumference
  of the orbit
• It takes longer for the trip
• Also, gravity weakens by inverse square law the
  farther out you go
• Make the distance so that it takes 24 hours for
  the orbit
• Satellite is stationary in the sky!!!

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The Moon

• Distance is about 240,000 miles
• Takes 27.3 days to make an orbit

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Elliptical Orbits

• Give the object a speed a bit greater than 8
  km/sec and the orbit will be elliptical

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Elliptical Orbits
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Elliptical Orbits
Sum of distances from foci to point on the
ellipse is a constant!!!
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Keplers Laws

• Each planet moves in an elliptical orbit with the
  sun at one focus of the ellipse.
• The line from the sun to any planet sweeps out
  equal areas of space in equal time intervals
• The squares of the periods of the planets are
  proportional to the cubes of their average
  distances from the sun

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Keplers Laws
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Energy Conservation
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Energy Conservation
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Energy Conservation
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Escape Speed
Throw at gt 11.2 km/sec and ball escapes the pull
of the earth!!!