Motion in Two and Three Dimensions

Distance Distance - How far you actually traveled. Displacement - Change in your position. –This is a vector and direction is important. Ex. You travel from Dayton, to Indianapolis, to Columbus. When your trip is done, what is your distance traveled and your displacement? 170 km 105 km Columbus Indianapolis Dayton

Position Vector Scalar - Magnitude (Size) Vector - Magnitude and Direction Position Vector - Location relative to an origin. 3.5 m 25° x y

Vector Addition Place vectors tail to head. Sum is from the tail of the first to the head of the last vector. A B A B C Solve Graphically

Vector Subtraction Same as adding a negative. -1 changes the vector’s direction by 180°. A C A B C Solve Graphically -A

Additional Properties Multiplication of a vector by a scalar –Can change the length of the vector. –Can change the sign of the vector. Algebraic Properties of Vectors –Commutative –Associative –Distributive

Law of Sines & Cosines Can perform vector addition using the laws of sines and cosines. A B C ° Law of CosinesLaw of Sines   

Coordinate Systems Project the vector on to the axis of the coordinate system. Ordered Pair of coordinates A AYAY AXAX  Convert back to polar

Unit Vectors Chose a vector of length one in the direction of each axis of the coordinate system. j ^ i ^ x y Ordered Pair becomes

Vector Addition (Again) Break each vector into components. A B C AxAx AyAy BxBx ByBy CxCx CyCy Add each set of components together.

A tracking station picks up the Aurora at a location 3.1 seconds later it is located at What is the magnitude of the displacement? What is its average velocity? The Aurora

Ferris Wheel You are located on a moving Ferris Wheel at King’s Island. Which of the following describes your motion. A) You are stationary. B) You are moving in a straight line. C) You are moving in a circle. D) You are moving in little loops around a larger circle.

Position, Velocity, Acceleration Position Velocity Acceleration Instantaneous

Acceleration and Velocity Constant Acceleration Ex. A rocket is traveling at a velocity of when its engines fail. What is its velocity after 20 s?

Acceleration and Displacement Integrating again gives Ex. A rocket is traveling at a velocity of when its engines fail. What is its displacement after 20s?

Galilean Transform Galilean transform is used when comparing velocities between two reference frames. (At least one is moving.) x’ y’ x y v v’ V O’ O O - Stationary Frame O’ - Moving Frame P or

At Sea Ex. A ship leaves Miami traveling due east at 6.00 m/s. It crosses the Gulf Stream, which is running at 1.79 m/s  75°. In what direction and at what speed does the ship travel with respect to Miami?

2-Dimensional Problems When solving 2-D problems, how many variables can there be? Initial vertical position Initial horizontal position Initial speed Initial angle of speed Horizontal acceleration Vertical acceleration Initial time Final vertical position Final horizontal position Final speed Final angle of motion Final time What is the minimum you need?

Projectile Motion Initial Velocity Acceleration X-component of displacement (x 0 =0) Y-component of displacement (y 0 =0)

Catapult A catapult is located in a castle that is 50m above the surrounding terrain. At what velocity must the catapult launch an object in order to hit a location 860m away if the launch angle is 50°?

Centripetal Acceleration Centripetal Acceleration - Object moves in a circle at constant speed. Acceleration  velocity

Rounding the Curve While driving at 24.5 m/s, you round a turn with a radius of curvature of 120 m. What is your acceleration? What direction is it? r v

Non-Uniform Circular Motion Radial Acceleration - Perpendicular to the velocity. Radially inward towards the center of the circular path. Tangential Acceleration - Parallel to the velocity. Slowing down and speeding up. arar atat arar atat Speeding Up Slowing Down

Ball on a String Each dot represents the position of a spinning object in equal time intervals. Indicate the acceleration at each dot.

Cedarville 500 Identify tangential acceleration & deceleration Identify zero & high radial acceleration

Under Siege A cannon is fired at an angle of 55° with a muzzle velocity of 300 m/s. The shell hits a castle which is at a 100m higher elevation. How long do the residents have to take cover?