General Physics I Vectors & 2D Motion Vectors In 1 dimension, we could specify direction with a + or - sign. In 2 (or 3) dimensions, we need more than a sign to specify the direction of something: To illustrate this, consider the position vector r in 2 dimensions. Example Where is Prestonsburg? Example: Choose origin at Pikeville Choose coordinates of distance (miles), and direction (N,S,E,W) In this case r is a vector that points 25 miles north. Page 1 Prestonsburg r Pikeville Vectors... There are two common ways of indicating that something is a vector quantity: Boldface notation: A A= A “Arrow” notation: A Vectors... The components of r are its (x,y) coordinates r = (rx ,ry ) = (x,y) We will consider this in 2-D: rx = x = r cos θ where r = |rr | ry = y = r sin θ (x,y) y θ = arctan (y / x) = tan-1 (y / x) r θ x Page 2 Vectors... The magnitude (length) of r is found using the Pythagorean theorem: r y r = r = x2 + y 2 x The length of a vector clearly does not depend on its direction. Vector addition: Consider the vectors A and B. Find A + B. A A B A B C=A+B B We can arrange the vectors as we want, as long as we maintain their length and direction!! Page 3 Vector addition using components: Consider C = A + B. Comparing components of (a) and (b): Cx = Ax + Bx Cy = Ay + By By C B A θ Ax Page 4 Bx Ay For Triangle A: We would find a similar solution for Triangle B. Therefore: Cx = Ax + Bx = A cos q + B cos q Cy = Ay + By = A sin q + B sin q C = Cx 2 + Cy 2 & q= tan-1 (Cy / Cx) For Example... A = 10 m, B = 20 m, and C = 7 m Page 5 Solution... 2D Kinematics We will deal with 2-D problems when acceleration is constant: Choose y axis to be along direction of acceleration Choose x axis to be along the “other” direction of motion Example Throwing a baseball (neglecting air resistance) Example: Acceleration is constant (gravity) Choose y axis up: ay = -g Choose x axis along the ground in the direction of the throw Page 6 “x” and “y” components of motion are independent. A man on a train tosses a ball straight up in the air. View this from two reference frames: Reference frame on the moving train. Reference frame on the ground. Velocity Components Page 7 The “x” & “y” Velocity Vectors Page 8 Problem: Fall Time Recall that there is no “gravity” in the xdirection ! So…Which kid hits the water first? Problem: Sammy’s Homer?? Sammy Sosa clobbers a fastball toward center-field. The ball is hit 1 m (yo ) above the plate, and its initial velocity is 36.5 m/s (vo) at an angle of 30o (θ θ) above horizontal. The center-field wall is 113 m (D) from the plate and is 3 m (h) high. What time does the ball reach the fence? Does Sammy get a home run? v y0 h θ D Page 9 Solution... Shooting the Monkey (tranquilizer gun) Where does the zookeeper aim if he wants to hit the monkey? ( He knows the monkey will let go as soon as he shoots ! ) Page 10 Shooting the Monkey... If there were no gravity, simply aim at the monkey End of Vectors & 2D Motion Lecture Page 11