Topic 1: 2D Motion PHYSICS 231
Current Assignments Homework Set 1 due this Thursday, Jan 20, 11 pm Homework Set 2 due Thursday, Jan 27, 11pm Reading: Chapter 4,5 for next week 2/1/11 Physics 231 Spring 2011 2
Key Concepts: 2D motion Vector. Displacement, Velocity, and Acceleration are vectors. Representations: Graphical, angle and magnitude, components. Add, subtract vectors. Decompose in components, and add components to obtain total vector. Treat 2D motion as 2 separate problems, one for x-direction and one for y-direction. (decompose all vectors in x- and y- components) 2/1/11 Physics 231 Spring 2011 3
Clicker Quiz What is a vector A: A set of coordinates B: A quantity that has a magnitude and a direction C: A pointer that indicates a direction D: An arrow 2/1/11 Physics 231 Spring 2011 4
Clicker Quiz If A is a vector, A x denotes A: The x-component B: The magnitude in x-direction C: The length D: The magnitude of the vector. 2/1/11 Physics 231 Spring 2011 5
Vector Concept A vector is a quantity that has a direction and a magnitude Examples so far: displacement, velocity, and acceleration In 1D the direction was given by the sign and a coordinate axis direction. In 2D its more complicated. Different representations can be used 2/1/11 Physics 231 Spring 2011 6
Representation: Graphical Graphically as arrow: Direction is direction of arrow Magnitude is length of arrow (need to define some scale) Note: where one draws the arrow is irrelevant. Addition by adding arrows (form a chain) Subtraction by adding a inverted arrow 2/1/11 Physics 231 Spring 2011 7
Clicker Quiz Which 2 vectors are the same? A B C D: They are all different vectors 2/1/11 Physics 231 Spring 2011 8
Vector addition/subtraction River, water velocity: V r + V b v b v r Velocity of boat relative to the water 2/1/11 Physics 231 Spring 2011 9
Checking Understanding Which of the vectors below best represents the vector sum P + Q? Slide 3-16
Checking Understanding Which of the vectors below best represents the difference P Q? Slide 3-18
Checking Understanding Which of the vectors below best represents the difference Q P? Slide 3-20
Representation: Components Components: A = A x + A y Magnitudes A x and A y are called x- and y-components y A θ A y Parallel to y-axis A x Parallel to x-axis Calculate components for given A X A x =A cosθ, A y =A sinθ Calculate magnitude of A if components are given: 2/1/11 Physics 231 Spring 2011 13
Checking Understanding What are the x- and y-components of these vectors? A. 3, 2 B. 2, 3 C. -3, 2 D. 2, -3 E. -3, -2 Slide 3-23
Checking Understanding What are the x- and y-components of these vectors? A. 3, 2 B. 2, 3 C. 3, 4 D. 1, 1 E. 4, 4 Slide 3-23
Checking Understanding What are the x- and y-components of these vectors? A. 3, -1 B. 3, -4 C. -3, 3 D. 4, -3 E. -3, -4 Slide 3-25
2D Motion Treat as two separate 1D problems with time as the common parameter In X-direction (calculate with x-components): v x (t) = v 0x + a x t x(t) = x 0 + v 0x t + 1 2 a xt 2 In Y-direction (calculate with y-components): v y (t) = v 0y + a y t y(t) = y 0 + v 0y t + 1 2 a y t 2 2/1/11 Physics 231 Spring 2011 17
Vector algebra graphical Addition: A + B=C add beginning of B to end of A, C points then from beginning of A to end of B Negative vector: B=-A: B has opposite direction of A, but same magnitude (flip arrow) Subtraction: A B = C is same as A + (-B) = C (flip direction of B then add) Multiply with scalar s: B = A * s direction stays the same, multiply length with s (for s<0 flip direction) 2/1/11 Physics 231 Spring 2011 18
Vector algebra components In component notation things become simple: Addition: add components Subtraction: subtract components Multiply with scalar: multiply each component 2/1/11 Physics 231 Spring 2011 19
Clicker Quiz A cart is moving along a track when a ball is vertically launched off the moving cart. Will the ball: A: hit the track behind the cart B: hit the cart C: hit the track ahead of the cart 2/1/11 Physics 231 Spring 2011 20
Clicker quiz A spring mechanism on a moving (to the right) cart gives a ball an upward velocity. Which best represents the vector of the initial velocity of the ball just when it leaves the spring? Vertical y Vertical y Vertical y Vertical y Horizontal x Horizontal x Horizontal x Horizontal x A B C D 2/1/11 Physics 231 Spring 2011 21
Clicker Quiz The gun barrel is straight and aligned with the aim of the gun. The monkey is hanging on the tree but will let go exactly at the moment the bullet leaves the gun. Where does the hunter need to aim the gun at in order to hit the monkey? A: A bit above the monkey B: Right at the monkey C: A bit below the monkey 2/1/11 Physics 231 Spring 2011 22