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Chapter 5 Projectile motion

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1. Recall: a projectile is an object only acted upon by gravity

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2. Chapter 4: [linear motion] straight line motion that was ONLY vertical or ONLY horizontal motion

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3. Chapter 5: looks at motion that follows a diagonal path or a curved path

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4. When you throw a baseball, it travels in an curved path.

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5. We will separate curved motion into independent x and y motions

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6. vertical motion is not affected by the horizontal motion. And the horizontal motion is not affected by the vertical motion.

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7. Observe: a large ball bearing is dropped at the same time as a second ball bearing is fired horizontally.

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What happens?

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Remember adding 2 perpendicular vectors horizontal and vertical vectors.

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8. Remember: When we add perpendicular vectors we use Pythagorean theorem to find the resultant.

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Boat in a river

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9. a river is 120 meters wide and has a current of 8 m/sec.

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Traveling up and down stream

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10. How fast will a boat drift downstream?

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11. V total = V boat + V current V total = 0 + 8 = 8 m/sec

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Now using the motor…

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12. Suppose the motor moves the boat at 15 m/sec.

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12. Suppose the motor moves the boat at 15 m/sec. how fast will the boat travel downstream [with the current]?

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13. Using motor with current: V total = V boat + V current

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Total velocity traveling downstream: V total = V boat + V current = 15 + 8 = 23 m/sec

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14.Using motor against the current: total velocity traveling upstream [AGAINST the current] Vtotal = Vboat –Vcurrent

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15. Going upstream: Velocity of current and boat are opposite directions Vtotal = Vboat –Vcurrent

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Vtotal = 15 -8 = 7 m/sec

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Crossing the river

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16. If there was no current, how many seconds for the boat to travel 120 meters to reach the opposite side?

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Velocity = distance time or time = distance velocity

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time = distance velocity time = 120 m 15 m/sec time = 8 seconds

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But there if is a current, what happens when you try to go straight across the river from A to B?

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18. If there is a current, The boat still crosses in 8 seconds, but it lands downstream at point C

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19. Add the perpendicular velocity vectors add to find the resultant velocity

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The triangles are similar:

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20. In this example, Every second the boat travels 15 meter in the x direction IT ALSO TRAVELS 8 meter in the y direction

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How far down stream is the boat when it reaches the opposite shore?

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Velocity = distance time so distance = velocity X time distance = 8 m/sec X 8 sec = 64 m

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What if you want to travel straight across and land at B, not C?

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21) If you want to go from A to B instead, you must point the boat diagonally upstream to compensate for the current.

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22) Planes are affected by the wind the same way

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23) Head wind: slows the plane [opposite direction]

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24. Tail wind: speeds the plane up [same direction]

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25. crosswind: blows plane off course [wind perpendicular to direction of plane]

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Break Vboat into Vx andV y components

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Use pythagorean theorem to find Vx.

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V x 2 + V y 2 = V boat 2 V x 2 + 8 2 = 15 2 V x 2 =225-64 = 161 V x =12.7 m/sec

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How many seconds to cross? Velocity = Distance/time T = D/V

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How many seconds to cross? From A to B T = D/V T = 120/12.7 T = 9.4 sec

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PROJECTILE MOTION

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Projectile motion: A projectile that has horizontal motion has a parabolic trajectory We can separate the trajectory into x motion and y motion.

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In the x direction: constant velocity V x = constant distance in x direction X = V x t

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In y direction: free fall = constant acceleration. Velocity in y direction : V = V o – g t Distance in y direction Y = Y o + V o t – ½ g t 2

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The range of a projectile is the maximum horizontal distance.

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Range and maximum height depend on the initial elevation angle.

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If you throw a projectile straight up, the range = 0 height is maximum. 0 degrees : the minimum range but the maximum height.

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The maximum range occurs at elevation 45 o

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And for complementary angles 40 and 50 degrees 30 and 60 degrees 15 and 75 degrees 10 and 80 degrees

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The range is identical for complementary angles BUT the larger elevation angle gives a greater maximum height.

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The height of a projectile at any time along the path can be calculated.

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First calculate the height if there was no gravity. If that case, a projectile would follow a straight line path

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the projectile is always a distance 5t 2 below this line.

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Y = v oy t – ½ gt 2 Y = v oy t – 5t 2 i

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summary

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Vectors have magnitude and direction Scalars have only magnitude

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The resultant of 2 perpendicular vectors is the diagonal of a rectangle that has the 2 vectors as the sides.

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The perpendicular components of a vector are independent of each other.

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The motion of a boat in a stream is the sum of a constant velocity of a boat [x dir] and the constant velocity of the stream [y dir]

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The path of a boat crossing a stream is diagonal

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The horizontal component of a projectile is constant, like a ball rolling on a surface with zero friction. Objects in motion remain in motion at constant speed.

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The vertical component of a projectile is same as for an object in free fall.

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The vertical motion of a horizontally fired projectile is the same as free fall.

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For a projectile fired at an angle, the projectile will be 5t 2 below where it would be if there was no gravity.

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