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Remember:. Problems worthy of attack Prove their worth by hitting back --Piet Hein. Learning physics involves resistance training!. II. 2-D (and 3-D) Motion. Definition and Properties of Vectors Position, Velocity and Acceleration Vectors 2-D Motion with Constant Acceleration

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  1. Remember: Problems worthy of attack Prove their worth by hitting back --Piet Hein Learning physics involves resistance training!

  2. II. 2-D (and 3-D) Motion • Definition and Properties of Vectors • Position, Velocity and Acceleration Vectors • 2-D Motion with Constant Acceleration • Projectile Motion

  3. A. Definition and Properties of Vectors A vector is a mathematical representation of a physical quantity that has both magnitude and direction. A quantity with magnitude but no direction is called a scalar. Scalars: mass, temperature, volume Vectors: velocity, acceleration, force, momentum

  4. Graphical Depiction of a Vector Direction is obvious from sketch, while length of arrow is proportional to vector magnitude

  5. Analytical Depiction of a Vector y Ay A  Ax x Ax and Ay are called the x- and y-components of the vector A. It is written in terms of the unit vectors i and j, which are vectors along the coordinate axes with length equal to 1.

  6. Vector Addition and Subtraction Graphical Method (qualitative): “Head-to-Tail” Analytical Method (quantitative)

  7. B. Position, Velocity, and Acceleration Vectors General position vector: General velocity vector: where:

  8. B. Position, Velocity, and Acceleration Vectors (contd.) General acceleration vector: where:

  9. C. 2-D (3-D) Motion with Constant Acceleration General expression: If acceleration is constant: or:

  10. In any vector equality, the components must be equal, so by inspection we find: One more integration step shows that: And these equations combine to give:

  11. D. Projectile Motion Here we assume the special case where an object traveling in a plane near the earth’s surface experiences only the constant acceleration of gravity: So, for example, we have the kinematic equations: And so on…

  12. v0 = ? x0 = 0 y0 = 1.9 m x= 38.8 m y= 0 A baseball example! If a catcher throws a ball horizontally from home plate toward second base at an initial height of 1.9 m, what must the initial speed of the ball be so that it reaches the shortstop’s mitt at exactly ground level?

  13. y v0 x x0 = 0 y0 = 0 x= 150 ft y= 0 Ready, aim… At what angle above the horizontal must a gun be aimed if you want to hit a target 150 ft away at the same y-coordinate as the gun? The gun has a muzzle velocity of 1500 ft/s.

  14. v0   x0 = 0 y0 = 0 h v2 y= -h v1 ax = 0 ay = -g An analytical example (no numbers!) 3.122 Two balls are thrown with equal speeds from the top of a cliff of height h. One ball is thrown at an angle  above the horizontal, while the other is thrown at an angle  below the horizontal. Show that each ball strikes the ground with the same speed, and find that speed in terms of h and the initial speed v0.

  15. Finding a plan of attack 3.118 Suppose you can throw a ball a distance x0 when standing on level ground. How far can you throw it from a building of height h = x0 if you throw it at (a) 0°, (b) 30°, (c) 45°? •What am I asked to find? •What can I get from the given information? •What next? “how far can you throw it” = x – x0= x “you can throw a ball a distance x0” = maximum x maximum x over level ground implies a launch angle of 45° I know x and , so I can find an expression for v0 I have v0 and  and y, so I can find x for each part (in terms of x0).

  16. Remember: Problems worthy of attack Prove their worth by hitting back --Piet Hein

  17. And also: Subjects which disclose their full power, meaning, and beauty as soon as they are presented to the mind have very little of those qualities to disclose. --Charles Dutton (1882)

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