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.

Vectors... The components of any vector are its (x,y) coordinates We will consider this in 2-D for our r vector: y x  r Consider our right-triangle trigonometry: - So….. rx = the x-component = ry = the y-component =

Vectors... The magnitude (length) of r is found using the Pythagorean theorem (assuming we know rx & ry) The direction (angle) of r is found using the inverse tangent function (again, assuming we know rx & ry)

Vector addition: Cannot directly combine vectors that point in directions other than parallel and anti-parallel (opposite direction). 4m + 1m = 4m + 1m = Must use vector components if they are in different directions 4m + 1m =

Vector addition using components: Consider C = A + B. Comparing components of (a) and (b): Cx = Ax + Bx Cy = Ay + By C Bx A By B Ax Ay 

θ = tan-1 (Cy / Cx) For Triangle A: & Therefore: We would find a similar solution for Triangle B. Therefore: Cx = Ax + Bx = A cos θA + B cos θB Cy = Ay + By = A sin θA + B sin θB θ = tan-1 (Cy / Cx) &

For Example...

Solution...

“x” and “y” components of motion are independent. 2D Kinematics We will deal with 2-D problems when acceleration is constant: Choose y axis to be along direction of the acceleration due to gravity Choose x axis to be along the “other” direction of motion (no acceleration or deceleration in the horizontal direction) Example: Throwing a baseball (neglecting air resistance) Acceleration is constant (gravity) Choose y axis up: ay = -g Choose x axis along the ground in the direction of the throw “x” and “y” components of motion are independent.

Velocity Components

(ax = 0…no acceleration/deceleration in the x-direction) Using Vector Components with the Horizontal & Vertical Equations of Motion!! (ax = 0…no acceleration/deceleration in the x-direction) Total Flight Time:

The “x” & “y” Velocity Vectors

Problem: Fall Time Which set of binoculars hits the water first? Which set is moving fastest when it hits the water?

Problem: Frustrated With It All A disgruntled physics student, frustrated with finals, releases his tensions by bombarding the adjacent building, 18.4 m away, with water balloons. He fires one at 39o from the horizontal with an initial speed of 24 m/s. For how long is the balloon in the air? How far above the initial launch height does the balloon hit the opposing building?

Solution...

End of Vectors & 2D Motion Lecture