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13.4 - Section 14.4 Motion in Space Velocity and...

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Section 14.4 Motion in Space: Velocity and Acceleration “Applications of vector functions and vector derivatives” In Calc 1 and 2, we saw that derivatives and integrals were closely related to the concept of speed, distance traveled and acceleration. In this section, we shall generalize these ideas to vectors, tracking the motion of a body in 3-dimensional space (rather than the rather fake 2-d space developed in Calc 2). 1. Velocity, Acceleration and Force Vectors Suppose that vector r ( t ) is the position function for a particle P traveling through space. Then we define its velocity vector and acceleration vector as follows: Definition 1.1. The velocity (the rate of change of position with re- spect to time) of P at time t is defined to be vectorv ( t ) = lim h 0 vector r ( t + h ) - vector r ( t ) h = vector r ( t ) and the acceleration (the rate of change of velocity with respect to time) of P at time t is defined to be vectora ( t ) = lim h 0 vectorv ( t + h ) - vectorv ( t ) h = lim h 0 vector r ( t + h ) - vector r ( t ) h = vector r ′′ ( t ) . We define the speed of P at time t to be || vector r ( t ) || (this is the physical rate of change of distance with respect to time). Calculation of these values are straight forward. Example 1.2. Find the velocity, acceleration and speed of a particle whose position in space is given by vector r ( t ) = sin ( t ) vector i + t vector j + cos ( t ) vector k. We have vector r ( t ) = cos ( t ) vector i + vector j - sin ( t ) vector k , vector r ′′ ( t ) = - sin ( t ) vector i - cos ( t ) vector k , and || vector r ( t ) || = radicalbig (cos 2 ( t ) + 1 + sin 2 ( t )) = 2.
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