lecture-ch2 Motion Along a Straight Line

lecture-ch2 Motion Along a Straight Line - Contents 2...

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Unformatted text preview: Contents 2 Motion Along a Straight Line 1 2.1 Position and Displacement . . . . . . . . . . . . . . . . . . . . 1 2.2 Average Velocity and Average Speed . . . . . . . . . . . . . . 1 2.3 Instantaneous Velocity and Speed . . . . . . . . . . . . . . . . 1 2.4 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.5 Constant Acceleration . . . . . . . . . . . . . . . . . . . . . . 2 2.6 Differentiation Rules . . . . . . . . . . . . . . . . . . . . . . . 2 2.6.1 Taylor Series Expansion . . . . . . . . . . . . . . . . . 3 2.6.2 Exponential and Logarithmic Functions . . . . . . . . . 5 2.6.3 Trigonometric Functions . . . . . . . . . . . . . . . . . 6 2.7 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.7.1 Linear Motion . . . . . . . . . . . . . . . . . . . . . . . 12 2 Motion Along a Straight Line 2.1 Position and Displacement x x 1 x 2 Δ x A change from one position x 1 to another position x 2 is called a displace- ment Δ x , where Δ x = x 2 − x 1 2.2 Average Velocity and Average Speed v avg = Δ x Δ t = x 2 − x 1 t 2 − t 1 1 x Δ x Δ t t t 1 t 2 x 2 x 1 x ( t ) 2.3 Instantaneous Velocity and Speed v = lim Δ t → Δ x Δ t = dx dt 2.4 Acceleration Average Acceleration a avg = Δ v Δ t = v 2 − v 1 t 2 − t 1 Instantaneous Acceleration a = lim Δ t → Δ v Δ t = dv dt = d dt parenleftbigg dx dt parenrightbigg = d 2 x dt 2 2.5 Constant Acceleration When the acceleration is constant, we have a = a avg = v ( t ) − v (0) t − v ( t ) = v (0) + at which means v ( t ) increases linearly with time t . The displacement Δ x is equal to the average velocity time the time span. Thus x ( t ) − x (0) = v avg ( t − 0) 2 Now the velocity changes linearly with respect to time, so we have v avg = 1 2 ( v ( t ) + v (0)) = v parenleftbigg t 2 parenrightbigg Consequently, x ( t ) = x (0) + tv parenleftbigg t 2 parenrightbigg = x (0) + t parenleftBig v (0) + a 2 t parenrightBig = x (0) + v (0) t + a 2 t 2 2.6 Differentiation Rules a, b are constants. n is an integer. From af ( t + Δ t ) + bg ( t + Δ t ) − af ( t ) − bg ( t ) Δ t = a f ( t + Δ t ) − f ( t ) Δ t + b g ( t + Δ t ) − bg ( t ) Δ t , we get d dt ( af ( t ) + bg ( t )) = a df ( t ) dt + b dg ( t ) dt Since f ( t + Δ t ) g ( t + Δ t ) − f ( t ) g ( t ) Δ t = f ( t + Δ t ) − f ( t ) Δ t g ( t + Δ t )+ f ( t ) g ( t + Δ t ) − g ( t ) Δ t , we have d dt ( f ( t ) g ( t )) = df ( t ) dt g ( t ) + f ( t ) dg ( t ) dt which leads to d dt ( f 1 ( t ) f 2 ( t ) ..f n ( t )) = df 1 ( t ) dt f 2 ( t ) ..f n ( t ) + f 1 ( t ) df 2 ( t ) dt f 3 ( t ) ..f n ( t ) + .. + f 1 ( t ) f 2 ( t ) ..f n − 1 ( t ) df n ( t ) dt d dt ( f ( t ) n ) = nf ( t ) n − 1 df ( t ) dt As an example, dt n dt = nt n − 1 dt dt = nt n − 1 Since df ( t ) dt = d dt parenleftBig f ( t ) 1 n parenrightBig n = n parenleftBig f ( t ) 1 n parenrightBig n − 1 d dt parenleftBig f ( t ) 1 n parenrightBig 3 we have d dt parenleftBig f ( t ) 1 n parenrightBig...
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This note was uploaded on 05/14/2011 for the course ECON 101 taught by Professor Asdaf during the Spring '11 term at Universidad de San Buenaventura Bogota.

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lecture-ch2 Motion Along a Straight Line - Contents 2...

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