Kinematics in One Dimension

1. Kinematics in One Dimension Chapter 2

2. Motion • Motion is defined as movement from one place to another. However, before we can say an object has moved, we have to define the frame of reference. • The frame of reference is going to be an object that you pick and relate to the motion of the object. • For example: Are you moving right now?

3. Motion • Remember that all motion is relative. • You can pick your frame of reference to make it easier on you. For example, a train…

4. Motion • Mechanics is the study of the motion of objects and is divided into two parts • Kinematics – how objects move • Dynamics – why objects move

5. Kinematics • We will be studying kinematics in one dimension. • This means we will be dealing with translational motion only.

6. Kinematics • When we are studying motion, there are 4 factors that we have to keep track of: • Time • Displacement • Velocity • Acceleration

7. Distance • Distance is the measure of how far an object has traveled. A distance is never negative and is denoted with the symbol x for movement in the horizontal distance and y for movement in the vertical distance. • Example: Running 100 m on a track.

8. Displacement • Displacement is another way to measure the distance covered but a displacement has a direction associated with it. It is also denoted with the symbol x (for horizontal) but the x has an arrow over it. • A displacement can be negative. • Example: Texas two-step.

9. Displacement is change in position. • Δx = x – x0

10. Velocity • Just measuring the change in position is only half of the story. To understand the motion of an object we must also know the velocity. • Velocity is how fast the object moves and the direction that the object moves. It is denoted by the symbol v with an arrow over it.

11. Velocity is the change in the displacement over the change in time. • v = Δx/Δt = (x-x0)/(t-t0) • What are the units of velocity?

12. Velocity vs. Speed • We often use the terms speed and velocity interchangeably but speed is how fast the object goes without relation to the direction. • Speed = distance/time • Velocity = displacement/time

13. Scalar vs. Vector • Velocity has a magnitude and a direction. It is a vector quantity. Displacement is also a vector quantity. • Speed, however, has only a magnitude. It is a scalar quantity. Time is also a scalar quantity. • Vectors are usually boldfaced and have a half arrow drawn over the top of them.

14. 2 Kinds of Velocity • Average velocity is the total displacement divided by the time. Example: road trips • Instantaneous velocity is the velocity that you are going as the change in time approaches zero. Example: speedometer

15. Problem • The position of a runner as a function of time is plotted as moving along the x axis of a coordinate system. During a 3.00 s time interval, the runner’s position changes from x1 = 50.0 m to x2 = 30.5 m. What is the runner’s average velocity?

16. Acceleration • Changes in velocity over time are known as accelerations. • As we speed up and slow down, we are constantly accelerating in a positive or negative direction.

17. Acceleration • a = Δv/Δt =(vf-vi)/Δt • Acceleration is the change of velocity over a period of time. With this in mind, what would be the units of acceleration?

18. A shuttle bus slows to a stop with an average acceleration of 1.8 m/s2 . How long does it take the bus to slow from 9.0 m/s to 0.0 m/s?

19. Just like velocity, acceleration has a magnitude and a direction. • In order to change acceleration, you have to change velocity. This means a change in direction or a change in magnitude.

20. Constant Acceleration • Constant acceleration occurs when the velocity changes by the same interval over the same time interval. • This also means that the displacement increases by the same interval. This does NOT mean that the intervals are the same! • The most famous acceleration is what?

21. Uniformly Accelerated Motion • When the acceleration is constant, there exists several well defined relationships between the velocity and the displacement.

22. The Equations • v = vo + at • vavg= ½ (v+v0)

23. Problem • Suppose a planner is designing an airport for small planes. One kind of airplane that might use this airport must reach a speed before takeoff of 100 km/h (27.8 m/s) and can accelerate at 2.0 m/s2. If the runway is 150m long, can this plane reach the proper speed to take off?

24. Problem Solving Strategy • Read it! • Draw a diagram! • Identify knowns and unknowns. • Identify the equation and solve it algebraically. • Calculate! • Check it!

25. Problem • A baseball pitcher throws a fastball with a speed of 44 m/s. Estimate the average acceleration of the ball during the throwing motion. It is observed that in throwing the baseball, the pitcher accelerates the ball through a displacement of about 3.5 m from behind the body to the point where it is released.

26. Free Fall • What falls faster, a crumpled up piece of paper or a bowling ball? • Neither falls faster. They fall at the same rate. Can you think why that would be? • They are both falling with the same acceleration – gravity!!

27. Anything that falls has a constant acceleration of 9.8 m/s2. • This motion is known as free fall. • Free fall allows objects to fall at the same rate.

28. The time is the same for each ball. What happens to the distance covered? So what is changing?

29. Gravity is a downward directed acceleration. • What sign does it have? • Now, the acceleration due to gravity does change as you get higher from the ground, but it is negligible.

30. What would happen if you first threw a ball up? • The acceleration would still be 9.8 m/s2. If the movement is in the y plane, the acceleration is going to be due to gravity.

31. Free Fall • Watch the ball. What happens? • When you throw the ball in the air, it will slow down until it reaches the top. At the top of its path, it will stop for a short time. After it stops, it will speed up until it hits the ground.

32. Bob throws a ball upward with a velocity of 3 m/s. He catches it in his hand a short while later. • What is the velocity that he catches it at? • How long does it take to get to the top of its path? • How long does it take for him to catch the ball again?

33. Graphing

34. Graphing • We can visualize how an object moves by graphing it. • Remember that the independent variable goes on the x-axis and the dependent variable goes on the y-axis. • What is the independent variable in a D vs T graph?

35. D vs T • When we graph distance vs. time, we can identify several key parts: • The speed • The total distance • The total time

36. D vs T • The speed of the graph is demonstrated by the slope of the line. • The slope is defined as the change in the rise over the change in the run. • What is the rise in a D vs T graph? • What is the run in a D vs T graph?

37. D vs T • How do you think that you could figure out the total distance covered? • The last number on the line! • How do you think that you could figure out the total time? • The last number on the line! • How do you think that you could figure out the total displacement covered? • The difference from zero.

38. D vs T

39. D vs T

40. D vs T Summary • Flat line = no movement! • Slope = speed • Total distance = the last number • Total time = the last number • Total displacement = the difference from zero

41. V vs T • In a V vs T graph we can establish the same points as in a D vs T graph. However, we have to do it differently. • The velocity • The total distance • The total displacement • The total time • The acceleration

42. V vs T

43. V vs T Summary • Flat line = constant velocity • Slope = acceleration • Area under the curve is the distance • Area under the curve is the displacement but you need to consider direction. • Total time = the last time reading

44. What if the object is accelerating? Does this change the graph? • Yes it does! The D vs T graph becomes curved. • Can you figure out the velocity at a specific time? • Yes! You can take the tangent at that specific point.

45. When the object is accelerating in the positive direction, the curve is downward – like in a positive parabola. • When the object is accelerating in the negative direction, the curve is upward – like in a negative parabola.

46. Your turn!!! • Graph the following description of motion: • Bob decides that he wants to go to his friend Tim’s house. He walks 7 m to Tim’s house in 10 seconds. He then stays at Tim’s house for 1 minute. They decide to go to the comic book store 20 m away. It takes them 45 seconds to get there. They then spend 20 minutes there. • Graph the first 5 minutes of their journey.

47. What type of graph is it? • What was Bob’s speed for the first 10 seconds? • What was Bob and Tim’s speed for the trip to the comic book store? • What was their average speed?

48. Your turn again!!! • Graph the following description of motion: • Bob bikes over to his friend Fred’s house at a constant velocity of 2 m/s in 25 seconds. He stops at Fred’s house for 45 seconds and then they race to school at an acceleration of 0.5 m/s2 for 1 minute. They then coast to a stop for 30 seconds.

49. What type of graph was it? • What was their acceleration for the last 30 seconds of their trip? • What was the total displacement covered?

50. Translating the Graphs • If you know one of the graphs, you can get any of the others. • This is known as translation.