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2D Motion - Chapter 4 Motion in Two Dimensions Chapter...

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1 Chapter 4: Chapter 4: Motion in Two Dimensions Position and Displacement Position and Displacement circle6 The position of an object is described by its position vector, r circle6 The displacement of the object is defined as the change in its position barb4left Δ r = r f - r i circle6 The average velocity is the ratio of the displacement to the time interval for the displacement barb4left The direction of the average velocity is the direction of the displacement vector, Δ r Average Velocity t Δ Δ = r v
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2 Instantaneous Velocity Instantaneous Velocity circle6 The instantaneous velocity is the limit of the average velocity as Δ t approaches zero 0 lim t d t dt Δ → Δ = Δ r r v circle6 The direction of the instantaneous velocity vector at any point in a particle’s path is along a line tangent to the path at that point and in the direction of motion circle6 The magnitude of the instantaneous velocity vector is the speed barb4left The speed is a scalar quantity Average Acceleration Average Acceleration circle6 The average acceleration of a particle as it moves is defined as the change in the instantaneous velocity vector divided by the time interval during which that change occurs. f i f i t t t Δ = = Δ v v v a circle6 As a particle moves, Δ v can be found in different ways circle6 The average acceleration is a vector quantity directed along Δ v circle6 The instantaneous acceleration is the limit of the average acceleration as Δ v / Δ t approaches zero 0 lim t d t dt Δ → Δ = Δ v v a Instantaneous Acceleration
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3 Kinematic Equations for Two Kinematic Equations for Two- Dimensional Motion Dimensional Motion circle6 For constant acceleration, 2D equation of motion is similar to those of one-dimensional kinematics j y i x r arrowrightnosp + = Position, Velocity: j i j i arrowrightnosp arrowrightnosp y x v v dt dy dt dx dt r d v + = + = = a = constant, a x and a y are also constant We can use 1D kinematic equations t a v v t a v v y yi yf x xi xf + = + = and )t j i ( ) j i ( j ) ( i ) ( arrowrightnosp y x yi xi y yi x xi f a a v v t a v t a v v + + + = + + + = t a v v i f arrowrightnosp arrowrightnosp arrowrightnosp + = 2 2 2 1 and 2 1 t a t v y y t a t v x x y yi i f x xi i f + + = + + = 2 2 2 ) j i ( 2 1 ) j i ( ) j i ( j ) 2 1 ( i ) 2 1 ( j i t a a t v v y x t a t v y t a t v x y x r y x yi xi i i y yi i x xi i f f f arrowrightnosp + + + + + = + + + + + = + = 2 2 1 t a t v r r i i f arrowrightnosp arrowrightnosp arrowrightnosp arrowrightnosp + + = Similarly: is the vector sum of (i) the original position , (ii) displacement arising from the initial velocity (iii) and a displacement resulting from the constant acceleration f r arrowrightnosp t v i arrowrightnosp 2 2 1 t a arrowrightnosp i r arrowrightnosp
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4 Kinematic Equations, Component Equations Kinematic Equations, Component Equations circle6 v f = v i + a t
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