Velocities Corresponding to Sample Position Functions

Velocities Corresponding to Sample Position Functions - 2...

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Velocities Corresponding to Sample Position Functions Since we know that v ( t ) = x' ( t ) , we can now use our new knowledge of derivatives to compute the velocities for some basic position functions: for x ( t ) = c , c a c onstant, v ( t ) = 0 (using (F2)) for x ( t ) = at 2 + vt + c , v ( t ) = at + v (using (F1),(F2),(P1), and (P2)) for x ( t ) = cos wt , v ( t ) = - w sin wt (using (F3a)) for x ( t ) = vt + c , v ( t ) = v (using (F1),(P2)) Notice that in this last case, the velocity is constant and equal to the coefficient of t in the original position function! (4) is popularly known as "distance equals rate × time." Acceleration in One Dimension Just as velocity is given by the change in position per unit time, acceleration is defined as the change in velocity per unit time, and is hence usually given in units such as m/s 2 (meters per second 2 ; do not be bothered by what a second
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Unformatted text preview: 2 is, since these units are to be interpreted as (m/s)/s--i.e. units of velocity per second.) From our past experience with the velocity function, we can now immediately write by analogy: a ( t ) = v' ( t ) , where a is the acceleration function and v is the velocity function. Recalling that v , in turn, is the time derivative of the position function x , we find that a ( t ) = x'' ( t ) . To compute the acceleration functions corresponding to different velocity or position functions, we repeat the same process illustrated above for finding velocity. For instance, in the case x ( t ) = at 2 + vt + c , v ( t ) = at + v , we find a ( t ) = v' ( t ) = a ! (This suggests some method to the seeming arbitrariness of writing the coefficient of t 2 in the equation for x ( t ) as a .)...
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This note was uploaded on 02/09/2012 for the course PHY PHY2053 taught by Professor Davidjudd during the Fall '10 term at Broward College.

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