Chapter 4 Motion in Two and Three Dimensions

1. Chapter 4 Motion in Two and Three Dimensions

2. Position The position of an object is described by its position vector,

3. Displacement The displacement of the object is defined as the change in its position,

4. Velocity • Average velocity • Instantaneous velocity

5. Instantaneous velocity Vector of instantaneous velocity is always tangential to the object’s path at the object’s position

6. Acceleration • Average acceleration • Instantaneous acceleration

7. Acceleration • Acceleration – the rate of change of velocity (vector) • The magnitude of the velocity (the speed) can change – tangential acceleration • The direction of the velocity can change – radial acceleration • Both the magnitude and the direction can change

8. Chapter 4 Problem 7 A particle starts from the origin with velocity 5 i m/s at t = 0 and moves in the plane with a varying acceleration given by a = (6 t1/2 j) m/s2, where t is in s. (a) Determine the vector velocity of the particle as a function of time. (b) Determine the position of the particle as a function of time.

9. Projectile motion • A special case of 2D motion • An object moves in the presence of Earth’s gravity • We neglect the air friction and the rotation of the Earth • As a result, the object moves in a vertical plane and follows a parabolic path • The x and y directions of motion are treated independently

10. Projectile motion – X direction • A uniform motion: ax = 0 • Initial velocity is • Displacement in the x direction is described as

11. Projectile motion – Y direction • Motion with a constant acceleration: ay = – g • Initial velocity is • Therefore • Displacement in the y direction is described as

12. Projectile motion: putting X and Y together

13. Projectile motion: trajectoryand range

14. Projectile motion: trajectoryand range

15. Chapter 4 Problem 15 A ball is tossed from an upper-story window of a building. The ball is given an initial velocity of 8.00 m/s at an angle of 20.0° below the horizontal. It strikes the ground 3.00 s later. (a) How far horizontally from the base of the building does the ball strike the ground? (b) Find the height from which the ball was thrown. (c) How long does it take the ball to reach a point 10.0 m below the level of launching?

16. Uniform circular motion • A special case of 2D motion • An object moves around a circle at a constant speed • Period – time to make one full revolution • The x and y directions of motion are treated independently

17. Uniform circular motion • Velocity vector is tangential to the path • From the diagram • Using • We obtain

18. Centripetal acceleration

19. Centripetal acceleration • During a uniform circular motion: • the speed is constant • the velocity is changing due to centripetal(“center seeking”) acceleration • centripetal acceleration is constant in magnitude (v2/r), is normal to the velocity vector, and points radially inward

20. Relative motion • Reference frame: physical object and a coordinate system attached to it • Reference frames can move relative to each other • We can measure displacements, velocities, accelerations, etc. separately in different reference frames

21. Relative motion • If reference frames A and B move relative to each other with a constant velocity • Then • Acceleration measured in both reference frames will be the same

22. Answers to the even-numbered problems • Chapter 4 • Problem 6: • v = - 12.0 t j^ m/s; a = - 12.0 j^ m/s2 • r = (3.00 i^ - 6.00 j^) m; v = - 12.0 j^ m/s

23. Answers to the even-numbered problems Chapter 4 Problem 26: 0.281 rev/s

24. Answers to the even-numbered problems Chapter 4 Problem 30: (b) 29.7 m/s2 (c) 6.67 m/s at 36.9° above the horizontal

25. Answers to the even-numbered problems Chapter 4 Problem 36: 18.0 s