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Particle Acceleration and Velocity Analysis

This text explains the analysis of particle acceleration and velocity, including finding the highest point, graph analysis, and direction of movement.

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Particle Acceleration and Velocity Analysis

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  1. The acceleration is the derivative of the velocity. a(t) = -et 1 + e2t a(2) = - e2 1 + e4 Part (a) v(t) = 1 - tan-1(et) a(2) = -.132 or -.133

  2. Part (b) v(2) = 1 - tan-1(e2) v(2) = -.436 a(2) = -.133 from Part (a) Since both v(2) and a(2) are negative, the speed is increasing. Remember from earlier in the year that speed INCREASES when v(t) and a(t) have the SAME sign, and speed DECREASES when the signs are OPPOSITE.

  3. Part (c) The particle reaches its highest point when v = 0. v(t) = 1 - tan-1(et) 0 = 1 - tan-1(et) tan-1(et) = 1 tan1 = et et=tan1 ln (et) =ln (tan1)

  4. Radians required here!!! Part (c) t (ln e) =ln (tan1) t =ln (tan1) t =.443 sec ln (et) =ln (tan1)

  5. Part (c) To justify that t=.443 is the highest point, we’ll find the velocity somewhere below and above t=.443 seconds. =.067 v(.3) = 1 - tan-1(e.3) t =.443 sec = -.026 v(.5) = 1 - tan-1(e.5) Since v(t) is positive for t<.443 and v(t) is negative for t>.443, the particle’s graph has an absolute maximum at t=.443

  6. Integrate both sides y(2) = -1 + [1- tan-1(et)] dt 2 0 Part (d) v(t) = 1 - tan-1(et) y(2) = -1.360 or -1.361 The particle is moving AWAY from the origin since v(2) and y(2) are both negative.

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