2. Motion in One Dimension Physics Lecture Notes dx dt x t h h/ 2 g Motion in One Dimension

3. We will devote the next several months to learning about the As we focus on the language, principles, and laws which describe and explain the motion of objects, your efforts should center around internalizing the meaning of the information. Avoid memorizing the information; and avoid abstracting the information from the physical world which it describes and explains. Rather, contemplate the information, thinking about its meaning thinking about its meaning,and its applications. physics of motion.

4. Kinematics: is the science of describing the motion of objects using words,diagrams,numbers,graphs, and equations.

5. Motion in 1 Dimension v  In the study of kinematics, we consider a moving object as a particle. A particle is a point-like mass having infinitesimal size and a finite mass. Motion in One Dimension

6. 0246 22 44 66 x Displacement Motion in 1 Dimension The displacement of a particle is defined as its change in position. (m)  x = x f  x i = 5 m  0 m = 5 m Note: Motion to the right is positive Motion in One Dimension

7. 0246 22 44 66 x Displacement Motion in 1 Dimension The displacement of a particle is defined as its change in position. (m)  x = x f  x i =  6 m  5 m =  11 m Note: Motion to the left is negative Motion in One Dimension

8. 0246 22 44 66 x Displacement Motion in 1 Dimension The displacement of a particle is defined as its change in position. (m)  x = x f  x i = (  1 m)  (  6 m) = 5 m Note: Motion to the right is positive Motion in One Dimension

9. Motion in 1 Dimension Average velocity The average velocity of a particle is defined as x t x1x1 x2x2 t1t1 t2t2 xx tt Velocity is represented by the slope on a displacement-time graph Motion in One Dimension

10. Motion in 1 Dimension Average speed The average speed of a particle is defined as Motion in One Dimension

11. Motion in 1 Dimension Instantaneous velocity The instantaneous velocity v, is defined as the limiting value of the ratio xx tt x t Instantaneous velocity is represented by the slope of a displacement-time graph Motion in One Dimension

12. Motion in 1 Dimension Instantaneous speed The instantaneous speed of a particle is defined as the magnitude of its instantaneous velocity. Motion in One Dimension

13. Motion in 1 Dimension Average acceleration The average acceleration of a particle is defined as the change in velocity  v x divided by the time interval  t during which that change occurred. v t v1v1 v2v2 t1t1 t2t2 vv tt Acceleration is represented by the slope on a velocity-time graph Motion in One Dimension

14. Motion in 1 Dimension Instantaneous acceleration The instantaneous acceleration a, is defined as the limiting value of the ratio vv tt v t Instantaneous acceleration is represented by the slope of a velocity-time graph Motion in One Dimension

15. Motion in 1 Dimension t x t v t a Displacement, velocity and acceleration graphs The slope of a velocity-time graph represents acceleration The slope of a displacement-time graph represents velocity Motion in One Dimension

16. Motion in 1 Dimension t x t v t a tt Displacement, velocity and acceleration graphs The area under an acceleration-time graph represents change in velocity. vv The area under a velocity-time graph represents displacement. xx Motion in One Dimension

17. Motion in 1 Dimension Definitions of velocity and acceleration Average velocity Average acceleration Motion in One Dimension

18. Motion in 1 Dimension For constant acceleration An object moving with an initial velocity v o undergoes a constant acceleration a for a time t. Find the final velocity. vovo v time = 0time = t Solution: Eq 1 Motion in One Dimension

19. Motion in 1 Dimension For constant acceleration An object moving with a velocity v o is passing position x o when it undergoes a constant acceleration a for a time t. Find the object’s final position. time = 0time = t xoxo x Solution: Eq 2 Motion in One Dimension

20. Motion in 1 Dimension Eq 1 Eq 2 Solve Eq 1 for a and sub into Eq 2: Solve Eq 1 for t and sub into Eq 2: Eq 3 Eq 4 Motion in One Dimension

21. Motion in 1 Dimension Eq 1 Eq 2 Eq 3 Eq 4 Equations for kinematics for constant acceleration Motion in One Dimension

22. Motion in 1 Dimension The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals (a) 0 to 2 s (b) 0 to 4 s (c) 2 s to 4 s (d) 4 s to 7 s (e) 0 to 8 s. Problem 2-3 Motion in One Dimension

23. Motion in 1 Dimension The displacement versus time for a certain particle moving along the x axis is shown in Figure P2.3. Find the average velocity in the time intervals (a) 0 to 2 s (b) 0 to 4 s (c) 2 s to 4 s (d) 4 s to 7 s (e) 0 to 8 s. Problem 2-3 Motion in One Dimension

24. Motion in 1 Dimension The displacement versus time for a certain particle moving along the x axis is shown in Figure P2.3. Find the average velocity in the time intervals (a) 0 to 2 s (b) 0 to 4 s (c) 2 s to 4 s (d) 4 s to 7 s (e) 0 to 8 s. Problem 2-3 Motion in One Dimension

25. Motion in 1 Dimension The displacement versus time for a certain particle moving along the x axis is shown in Figure P2.3. Find the average velocity in the time intervals (a) 0 to 2 s (b) 0 to 4 s (c) 2 s to 4 s (d) 4 s to 7 s (e) 0 to 8 s. Problem 2-3 Motion in One Dimension

26. Motion in 1 Dimension The displacement versus time for a certain particle moving along the x axis is shown in Figure P2.3. Find the average velocity in the time intervals (a) 0 to 2 s (b) 0 to 4 s (c) 2 s to 4 s (d) 4 s to 7 s (e) 0 to 8 s. Problem 2-3 Motion in One Dimension

27. Motion in 1 Dimension A car is approaching a hill at 30.0 m/s when its engine suddenly fails, just at the bottom of the hill. The car moves with a constant acceleration of  2.00 m/s 2 while coasting up the hill. (a) Write equations for the position along the slope and for the velocity as functions of time, taking x = 0 at the bottom of the hill, where v o = 30.0 m/s. Problem 2-3 Motion in One Dimension

28. Motion in 1 Dimension (b) Determine the maximum distance the car travels up the hill. A car is approaching a hill at 30.0 m/s when its engine suddenly fails, just at the bottom of the hill. The car moves with a constant acceleration of  2.00 m/s 2 while coasting up the hill. At the maximum distance v = 0 Problem 2-3 Motion in One Dimension

29. Motion in 1 Dimension A ball is dropped from rest from a height h above the ground. Another ball is thrown vertically upward from the ground at the instant the first ball is released. Determine the speed of the second ball if the two balls are to meet at a height h/2 above the ground. v2v2 V 1 = 0 h h/ 2 Motion in One Dimension

30. Motion in 1 Dimension v2v2 V 1 = 0 h h/ 2 The vertical position (y 1 ) of the falling ball Time for the ball to fall h/2 g x = y 1 x o = h v o = 0 a =  g Motion in One Dimension

31. Motion in 1 Dimension v2v2 V 1 = 0 h h/ 2 g The vertical position (y 2 ) of the second ball x = y 2 x o = 0 v o = v 2 a =  g During time t =, the second ball rises to a height h/2 Motion in One Dimension

32. Motion in 1 Dimension More Graphs Motion in One Dimension

33. 01231564-2-3-4-5-6 Motion in One Dimension

34. 01231564-2-3-4-5-6 Motion in One Dimension

35. 01231564-2-3-4-5-6 Motion in One Dimension

36. 0 1 2 3 1 5 6 4 -2 -3 -4 -5 -6 24681012 Motion in One Dimension

37. 0 1 2 3 1 5 6 4 -2 -3 -4 -5 -6 24681012 Motion in One Dimension

38. 0 1 2 3 1 5 6 4 -2 -3 -4 -5 -6 24681012 Motion in One Dimension

39. 24681012 0 1 2 3 1 5 6 4 -2 -3 -4 -5 -6 m s Motion in One Dimension

40. 24681012 0 2 6 4 -2 -4 -6 m s v (m/s) t (s) 4812 2 1 0 -2 -3 Motion in One Dimension

41. 24681012 0 2 6 4 -2 -4 -6 m s v (m/s) t (s) 12 2 1 0 -2 -3 +4 m -12 m +8 m 48 Motion in One Dimension

42. 1624128402028 (s) x 4 8 12 16 20 24 28 (m) 1234 t 5 Motion in One Dimension

43. (s) x 4 8 12 16 20 24 28 (m) 1234 t 5 12345 t (s) 2 4 6 8 10 v (m/s) Displacement 25 m Motion in One Dimension

44. END Motion in One Dimension Motion in One Dimension