Two and Three-Dimensional Motion in Physics Lecture

1. University Physics: Mechanics Ch4. TWO- AND THREE-DIMENSIONAL MOTION Lecture 4 Iksan Bukhori, M.Phil iksan.bukhori@president.ac.id • Original Presentation by Dr. –Ing Erwin Sitompul 2018

2. Solution of Homework 3: The Beetles 2nd run, ? 1st run, 1.6 m New location 2nd run, 0.8 m 1st run, 0.5 m Starting point

3. Solution of Homework 3: The Beetles → D → C New location → B → Starting point A Thus, the second run of the green beetle corresponds to the vector

4. N W E S → D Solution of Homework 3: The Beetles (a) The magnitude of the second run? (b) The direction of the second run? The direction of the second run is 79.09° south of due east or 10.91° east of due south.

5. Moving in Two and Three Dimensions • In this chapter we extends the material of the preceding chapters to two and three dimensions. • Position, velocity, and acceleration are again used, but they are now a little more complex because of the extra dimensions.

6. Position and Displacement • One general way of locating a particle is with a position vectorr, → • The coefficients x, z, and y give the particle’s location along the coordinate axes and relative to the origin. • The following figure shows a particle with position vector • In rectangular coordinates, the position is given by (–3 m, 2 m, 5 m).

7. Position and Displacement • As a particle moves, its position vector changes in a way that the vector always extends from the originto the particle. • If the position vector changes from r1to r2, then the particle’s displacement delta is: → →

8. Average Velocity and Instantaneous Velocity → • If a particle moves through a displacement Δrin a time interval Δt, then its average velocityvavg is: → • The equation above can be rewritten in vector components as:

9. Average Velocity and Instantaneous Velocity → • The particle’s instantaneous velocity v is the velocity of the particle at some instant. • The direction of instantaneous velocity of a particle is always tangent to the particle’s path at the particle’s position.

10. Average Velocity and Instantaneous Velocity • Writing the last equation in unit-vector form: • This equation can be simplified by rewriting it as: → where the scalar components of v are: • The next figure shows a velocity vector v and its scalar x and y components. Note that v is tangent to the particle’s path at the particle’s position. → →

11. Average Velocity and Instantaneous Velocity The figure below shows a circular path taken by a particle. If the instantaneous velocity of the particle at a certain time isv = 2i – 2j m/s, through which quadrant is the particle currently moving when it is traveling clockwise (b) counterclockwise ^ ^ → Firstquadrant Thirdquadrant (a) clockwise (b) counterclockwise

12. Average and Instantaneous Acceleration → → • When a particle’s velocity changes from v1 to v2 in a time interval Δt, its average acceleration aavg during Δt is: → → • If we shrink Δt to zero, then aavg approaches the instantaneous acceleration a ; that is: →

13. Average and Instantaneous Acceleration • We can rewrite the last equation as → where the scalar components of aare: Acceleration of a particle does not have to point along the path of the particle

14. Average and Instantaneous Acceleration ^ ^ → A particle with velocity v0 = –2i + 4j m/s at t = 0 undergoes a constant acceleration a of magnitude a = 3 m/s2 at an angle 130° from the positive direction of the x axis. What is the particle’s velocity v at t = 5 s? → → Solution: At t = 5 s, Thus, the particle’s velocity at t = 5 s is

15. Exercises → → → Compute A + B –C and express the result in magnitude-angle notation (polar coordinate). → → → Compute A + B + C and express the result in magnitude-angle notation (polar coordinate).

16. Exercises → → → A shopper at a supermarket follows the path indicated by vectors S, H, O, and P in the figure. Given that the vectors have magnitudes S = 51ft, H = 45ft, O = 35ft, and P = 13ft. For all his movement, the shopper requires 8.5 min. Find: → • the magnitude of his displacement; • the direction of his displacement; • the magnitude of his average velocity; • the direction of his average velocity; • his average speed. • Hint: Answer using “ft” and “s.”