Chapter_2 - Chapter 2 Motion Along a Straight Line In this...

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Chapter 2 Motion Along a Straight Line In this chapter we will study 1-dimensional kinematics i.e. how objects move along a straight line. The following parameters will be defined: Displacement Average velocity Average Speed Instantaneous velocity Average and instantaneous acceleration Special case: motion with constant acceleration --- we will develop the equations that give us the velocity and position at any time. Example: free fall motion Finally we will study a graphical integration method that can be used to analyze the motion when the acceleration is not constant (2-1)
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Focus of this chapter: Motion will be along a straight line We will assume that the moving objects are “ particles ” i.e. all parts of an object move in the same way. The causes of the motion will not be investigated. Consider an object moving along a straight line taken to be the x-axis. The object’s position at time t is described by its coordinate x(t) defined with respect to the origin O (x=0). The coordinate x can be positive or negative (2-2) x 1 =-2m x 2 = 3m
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Displacement. If an object moves from position x 1 to position x 2 , the change in position is described by the displacement For example if x 1 = 5 m and x 2 = 12 m then Δ x = 12 – 5 = 7 m. The positive sign of Δx indicates that the motion is along the positive x-direction If x 1 = 5 m and x 2 = 1 m then Δx = 1 – 5 = -4 m. The negative sign of Δx indicates that the motion is along the negative x-direction Displacement is a vector quantity that has both magnitude and direction. In this restricted one-dimensional motion the direction is described by the algebraic sign of Δx 2 1 x x x ∆ = - Note: The actual distance for a trip is irrelevant as far as the displacement is concerned For example an object moves from x 1 = 5 m to x 2 = 200 m and then back to x 2 = 5 m. The total distance covered is 2×195 = 390 m, the displacement Δ x = 0m (2-3) . . . O x 1 x 2 x -axis motion Δx
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Average Velocity We plot position x (t) as function of time t for a moving animal. We can get an idea of “ how fast ” the animal moves from one position x 1 at time t 1 to a new position x 2 at time t 2 by determining the average velocity between t 1 and t 2 . Here x 2 and x 1 are the positions x(t 2 ) and x(t 1 ), respectively. The time interval Δt is defined as : Δt = t 2 – t 1 The units of v avg are: m/s Example: x(0) = -5m, x(4)= 2m v avg = [2-(-5)]/(4-0) = 1.75 m/s 2 1 2 1 avg x x x v t t t - = = - (2-4)
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Graphical determination of v avg On an x versus t plot we can determine v avg from the slope of the straight line that connects point ( t 1 , x 1 ) with point ( t 2 , x 2 ). In the plot below, t 1 =1 s, and t 2 = 4 s. The corresponding positions are : x 1 = - 4 m and x 2 = 2 m 2 1 2 1 2 ( 4) 6 m 2 m/s 4 1 3 s avg x x v t t - - - = = = = - - Average Speed s avg The average speed is defined in terms of the total distance traveled in a time
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Chapter_2 - Chapter 2 Motion Along a Straight Line In this...

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