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LectureNotes03 - Copyright R Janow –Spring 2012 Physics...

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Unformatted text preview: Copyright R. Janow –Spring 2012 Physics 111 Lecture 03 Motion in Two Dimensions SJ 8th Ed.: Ch. 4.1 –4.4 • Position, velocity, acceleration vectors • Average & Instantaneous Velocity • Average & Instantaneous Acceleration • Two Dimensional Motion with Constant Acceleration (Kinematics) • Projectile Motion (Free Fall) • Uniform Circular Motion • Tangential and Radial Acceleration • Relative Velocity and Relative Acceleration 4.1 Position, Velocity and Acceleration Vectors 4.2 Two Dimensional Motion with Constant Acceleration 4.3 Projectile Motion 4.4 A particle in Uniform Circular Motion 4.5 Tangential and Radial Acceleration 4.6 Relative Velocity and Relative Acceleration Copyright R. Janow –Spring 2012 Motion in two and Three Dimensions Extend 1 dimensional kinematics to 2 D and 3 D n descriptio z y, x, or y x, x ⇒ •Kinematic quantities become 3 dimensional k ˆ z j ˆ y i ˆ x r x + + ≡ ⇒ r k ˆ v j ˆ v i ˆ v v v z y x + + ≡ ⇒ r k ˆ a j ˆ a i ˆ a a a z y x + + ≡ ⇒ r • Motions in the 3 perpendicular directions can be analyzed independently •Vectors needed to manipulate quantities • Constant acceleration Kinematic Equations hold component-wise for each dimension t a v v x x0 x + = 2 x 2 1 x0 t a t v x x + =- ) x (x 2a v v x 2 x0 2 x- + = t a v v y y0 y + = 2 y 2 1 y0 t a t v y y + =- ) y (y 2a v v y 2 y0 2 y- + = ⊕ ⊕ Same for z •In vector notation each equation is 3 separate ones for x, y, z t a v v r r r + = 2 2 1 t a t v r r r r v r + =- z y, ) x (x 2a v v for same & x 2 x0 2 x- + = Copyright R. Janow –Spring 2012 Position and Displacement A particle moves along its path as time increases r r r i f r r r- ≡ Δ Displacement Positions Trajectory: y = f(x) Parameterized by time j ˆ ] y [y i ˆ ] x [x j ˆ y i ˆ x r i f i f- +- = Δ + Δ = Δ r f i f f i i t t ) (t r r ) (t r r < ≡ ≡ r r r r always points from the origin to the particle’s location r r • Path does not show time dependence. • Slopes of (tangents to) x(t), y(t), z(t) graphs would show velocity components Average Velocity t r interval time nt displaceme v avg Δ Δ = ≡ r r Same direction as r r Δ Rate of change of position Copyright R. Janow –Spring 2012 Instantaneous Velocity j ˆ v i ˆ v dt r d t r Lim t ) v ( Lim t v y x avg + = ≡ Δ Δ → Δ = → Δ ≡ r r r r dt dy v dt dx v y x where ≡ ≡ x y v r parabolic path Free Fall Example is tangent to the path in x,y i.e, to a plot of y vsx v r slope v x = t x t v ) t ( x x = t y y(t) is parabolic 2 t g 2 1- t v- ) t ( y y0 = Components of are tangent to graphs of corresponding components of the motion, viz: v r v x is tangent to x(t) v y is tangent to y(t) Copyright R. Janow –Spring 2012 Displacement in 3 Dimensions k ˆ z j ˆ y i ˆ x r i i i i + + ≡ r k ˆ z j ˆ y i ˆ x r Δ + Δ + Δ = Δ r r r r i f r r r- ≡ Δ f r r r i r r r Δ k v j ˆ v i ˆ v t r Lim t ) v ( Lim t v z y x avg r r r r + + = Δ Δ → Δ = → Δ ≡ Copyright R. Janow –Spring 2012Copyright R....
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LectureNotes03 - Copyright R Janow –Spring 2012 Physics...

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