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 A.Definition and Properties of Vectors B.Position, Velocity and Acceleration Vectors C.2-D Motion with Constant Acceleration D.Projectile Motion

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. A. Definition and Properties of Vectors Scalars: mass, temperature, volume Vectors: velocity, acceleration, force, momentum

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

Analytical Depiction of a Vector A x and A y 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. y x AyAy AxAx  A

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

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

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

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

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:

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…

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? v 0 = ? x 0 = 0 y 0 = 1.9 m x = 38.8 m 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. v0v0 x 0 = 0 y 0 = 0 x = 150 ft y = 0 x y

An analytical example (no numbers!) 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 v 0. v0v0 x 0 = 0 y 0 = 0 y = -h a x = 0 a y = -g   v2v2 v1v1 h

Finding a plan of attack Suppose you can throw a ball a distance x 0 when standing on level ground. How far can you throw it from a building of height h = x 0 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 – x 0 =  x “you can throw a ball a distance x 0 ” = maximum  x maximum  x over level ground implies a launch angle of 45° I know  x and , so I can find an expression for v 0 I have v 0 and  and  y, so I can find  x for each part (in terms of x 0 ).

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

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)