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Chapter 3 Motion in 2 dimensions

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1) Displacement, velocity and acceleration displacement is the vector from initial to final position

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1) Displacement, velocity and acceleration average velocity

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1) Displacement, velocity and acceleration instantaneous velocity v can change even if v is constant v is tangent to the path

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1) Displacement, velocity and acceleration average acceleration not in general parallel to velocity

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1) Displacement, velocity and acceleration instantaneous acceleration

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1) Displacement, velocity and acceleration instantaneous acceleration - object speeding up in a straight line acceleration parallel to velocity - object at constant speed but changing direction acceleration perp. to velocity

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2) Equations of kinematics in 2d Superposition (Galileo): If an object is subjected to two separate influences, each producing a characteristic type of motion, it responds to each without modifying its response to the other. That is, we consider x and y motion separately

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2) Equations of kinematics in 2d A bullet fired vertically in a car moving with constant velocity, in the absence of air resistance (and ignoring Coriolis forces and the curvature of the earth), will fall back into the barrel of the gun. That is, the bullet’s x-velocity is not affected by the acceleration in the y-direction. vxvx vyvy vyvy vxvx

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2) Equations of kinematics in 2d That is, we can consider x and y motion separately

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2) Equations of kinematics in 2d That is, we can consider x and y motion separately

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Example x y

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3) Projectile Motion (no friction) a) Equations Consider horizontal (x) and vertical (y) motion separately (but with the same time) Horizontal motion: No acceleration ==> a x =0 Vertical motion: Acceleration due to gravity ==> a y = ±g - usual equations for constant acceleration

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3) Projectile Motion (no friction) Example: Falling care package x Find x. Step 1: Find t from vertical motion Step 2: Find x from horizontal motion

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3) Projectile Motion (no friction) Example: Falling care package x Step 1: Given a y, y, v 0y

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3) Projectile Motion (no friction) Example: Falling care package x Step 2:

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3) Projectile Motion (no friction) b) Nature of the motion: What is y(x)? Eliminate t from y(t) and x(t): x y y(x) is a parabola

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3) Projectile Motion (no friction) c) Cannonball physics Find (i) height, (ii) time-of-flight, (iii) range

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3) Projectile Motion (no friction) c) Cannonball physics (i)Height: Consider only y-motion: v 0y given, a y =-g known Third quantity from condition for max height: v y =0

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3) Projectile Motion (no friction) c) Cannonball physics (ii) Time-of-flight: Consider only y-motion: v 0y given, a y =-g known Third quantity from condition for end of flight; y=0

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3) Projectile Motion (no friction) c) Cannonball physics (iii) Range: Consider x-motion using time-of-flight: x=v 0x t

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