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Motion in Two and Three Dimensions

Motion in Two and Three Dimensions. 4-2 Position and Displacement. The position vector is typically used to indicate the location of a particle. The position vector is a vector that extends from a reference point (usually the origin) to the particle. The position vector

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Motion in Two and Three Dimensions

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  1. Motion in Two and Three Dimensions

  2. 4-2 Position and Displacement The position vector is typically used to indicate the location of a particle. The position vector is a vector that extends from a reference point (usually the origin) to the particle.

  3. The position vector for a particle is the vector sum of its vector components.

  4. In unit vector notation can be written as:

  5. The coefficients x, y, and z are the scalar components.

  6. The coefficients x, y, and z give the particle’s location along the coordinate axes and relative to the origin.

  7. The figure shows a particle with position vector:

  8. As a particle moves, its position vector changes in such a way that the position vector always extends to the particle from the reference point (origin).

  9. If the position vector changes from during a certain time interval, then the particle’s displacement during that time interval is:

  10. We can rewrite this displacement as:

  11. Sample Problem 4-1

  12. Sample Problem 4-1 In figure 4-2 the position vector for a particle is initially and then later it is

  13. What is the particle’s displacement from ?

  14. Sample Problem 4-2 A rabbit runs across a parking lot on which a set of coordinate axes has been drawn. The coordinates of the rabbit’s position as functions of time t are given by: x = -0.3t2 + 7.2t + 28 y = 0.22t2 - 9.1t + 30

  15. At t = 15 seconds, what is the rabbit’s position vector in unit-vector notation and as a magnitude and an angle?

  16. Graph the rabbit’s path for t = 0 to t = 25 s.

  17. Average Velocity and Instantaneous Velocity If a particle moves through a displacement in time interval D t, then its average velocity is:

  18. Written as vector components:

  19. The instantaneous velocity is the value that approaches in the limit as Dt shrinks to 0.

  20. The direction of the instantaneous velocity of a particle is always tangent to the particle’s path at the particle’s position.

  21. The velocity of a particle along with the scalar components of

  22. Sample Problem 4-3 For the rabbit in sample problem 4-2, find the velocity at time t = 15 s, in unit vector notation and as a magnitude and an angle.

  23. Average Acceleration and Instantaneous Acceleration When a particle’s velocity changes from to in a time interval Dt, its average acceleration a avg during Dt is:

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