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Lesson_Text_1.2 - 1 Lesson 1.2 Equations of Motion 1...

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1 Lesson 1.2 Equations of Motion 1. Uniform and Non-uniform velocity: 2.0m 2.0 m 2.0 m 2.0 m 2.0 m Fig. 1The displacement is the same in every 1 s interval t=0 t=1 t=2 t=3 t=4 t=5 t =1 t =1 t =1 t =1 t =1 Uniform velocity occurs when equal displacements happen in equal intervals of time. Fig. 1 above shows the positions of a rolling ball in intervals of 1 s. The displacement of the ball in every interval of t = 1 s is the same, namely 2.0 m. 1 2.0 Uniform velocity = 2.0 . 1.0 x m m s t s - = = The following table shows the velocity of the rolling ball as time increases. Time (s) 0 1 2 3 4 Velocity (m/s) 2 2 2 2 2 A graph of velocity against time is a straight line parallel to the time axis as shown in fig. 2 below. 0 1 2 3 4 5 1 2 time, s velocity, m/s Fig. 2 Uniform velocity
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2 When unequal displacements take place in equal intervals of time, the velocity is non-uniform. When the velocity is uniform, displacement = (uniform velocity) (time). In fig. 3 below OA represents the time during which the object is in motion and OC represents the uniform velocity. CB is the graph of velocity against time. OABC is a rectangle and the area of this rectangle is the area under the velocity-time graph. But (velocity)(time) = (OA)(OC) = area of the rectangle OABC. time, s velocity, m/s O A B C Fig. 3 Displacement is area under the velocity-time graph This means that displacement of a moving object is given by the area under the velocity-time graph. This is true even when the velocity is not uniform. Example 1 : A ball moves on a horizontal floor with a uniform velocity of 2.5 m.s -1 . What is its displacement in 10 s? Solution : Uniform velocity = 2.5 m.s -1 Time = 10 s Displacement = (uniform velocity)(time) = 2.5 m.s -1 . 10 s = 25 m 2. Acceleration: A moving object has an acceleration whenever its velocity changes. Change in velocity may be in magnitude, direction or both. Whenever the velocity changes in any form, acceleration occurs. Acceleration is defined as the rate of change of velocity. If the velocity of a moving object is v 1 at time t 1 and v 2 at time t 2 ,
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3 The change in velocity v = v 2 – v 1 and the change in time = t = t 2 – t 1 -1 2 2 1 1 2 1 2 1 change in velocity ( ) m.s Acceleration = . change in time ( ) s v v v v v m s t t t t t - - - = = = - - Notice here that the unit of acceleration is m.s -2 . Since acceleration is the rate of change of velocity, the instantaneous acceleration is the derivative of the velocity function with respect to time, and can be written as: 2 2 0 t v dv d x a Lim t dt dt ∆ → = = = Again, as acceleration is the rate of change of velocity, it is given by the slope of the velocity-time graph as shown in fig. 4 and fig. 5 below. time velocity v1 v2 v t Fig. 4 Acceleration is the slope of the velocity-time gra time velocity v t Fig. 5 The slope of velocity-time graph is negative.
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