1. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Physics I 95.141 LECTURE 2 9/8/10 HW1 Due This Friday by 5 p.m. www.masteringphysics.com

2. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 EXAM 0 Results If you got a 4 or less, you are not ready for this course!

3. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Exam 0 results from last year For scores of <40%, DWF rate was ~70%!!

4. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Outline Reference Frames and Displacement Average velocity Instantaneous velocity Acceleration Motion with constant acceleration Freely falling bodies

5. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Frames of Reference (Position) Physics is all about describing and predicting the world around us. If you think about how we describe objects, one of the first things which should come to mind is POSITION. But position means nothing unless you know what the position is in reference to! In this class, we will most usually base problems in a Cartesian coordinate system

6. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Frames of Reference (Motion) With a frame of reference, we can now describe an object’s motion. For 1 dimensional (1D) motion (motion in a straight line) we generally use the x-axis to describe the object’s position. (For falling bodies, we tend to describe position using the y- axis)

7. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Distance vs. Displacement Distance: the total distance traveled by an object along its path –Distance is a scalar Displacement: how far an object is from its starting point –Displacement is a vector –A vector has both magnitude and direction –We label vectors with an arrow :

8. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Distance vs. Displacement Distance: the total distance traveled by an object along its path –Distance is a scalar Displacement: how far an object is from its starting point –Displacement is a vector

9. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Average Speed and Velocity Displacement helps us describe the position of an object. Speed and Velocity tell us the motion of the object. Speed: How far an object travels in a given time interval speed ≠ velocity Velocity: Displacement of an object over time Velocity has both a magnitude and a direction. It is a vector.

10. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Average Velocity vs. Average Speed Average speed Example: A runner takes 30 seconds to complete the loop. What is the runner’s average speed and velocity? R=3m Average velocity

11. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Instantaneous Velocity Average velocity does not tell the whole story… For instance, if the MassPike decided to time your travel from I-93 (exit 24) to Framingham (exit 12), 23.2 miles, it could calculate your average velocity. The speed limit is 55mph, and it takes you 30 minutes.

12. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Position vs. Time Plots We can describe an object’s motion (in 1D) with a position vs. time plot.

13. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Average Velocity Mathematically, the average velocity is the change in position over the change in time. Graphically, average velocity is the slope of the line between two points on the plot.

14. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Instantaneous Velocity (Graphically) Graphically, instantaneous velocity is the slope of the x vs t plot at a single point Zoom in….

15. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Instantaneous Velocity Mathematically, the instantaneous velocity is the derivative of the position function For a function which gives position (x) as a function of time (t), we can find the function of velocity as a function of time by taking the derivative of the position function

16. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Example Problem a)Determine displacement between t 1 =2s and t 2 =5s b)Determine average velocity during this time c)Determine magnitude of instantaneous velocity at t 2 =5s.

17. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Example Problem c) Determine magnitude of instantaneous velocity at t 2 =5s.

18. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Average Acceleration Even knowing the velocity doesn’t tell us the entire story! Velocity can change with time: this is called acceleration! Average Acceleration:

19. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Example Problem Car accelerates from rest to 90km/hr in 5s. What is average acceleration? Units!

20. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Average Deceleration If the magnitude of an object’s velocity is decreasing, it is said to be decelerating. Deceleration and acceleration are calculated in exactly the same way

21. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Average Deceleration

22. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Average Deceleration Here we see the boat decelerate from 6 knots to 5 knots in about 45s. 1 knot ~ 0.51 m/s

23. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Instantaneous Acceleration Value of average acceleration as Δt  0. When we refer to “acceleration” in this class, we are talking about instantaneous acceleration.

24. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Example If we are given x(t), we can find both velocity and acceleration as a function of time (v(t), a(t)). a)What is v(t)? b)What is average acceleration from t 1 =1 to t 2 =2s? c)What is a(t)?

25. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 For Constant Acceleration ONLY Instantaneous and average acceleration are equal. Assume time interval goes from t=0s to t At t=0s, v=v o. Can find v(t):

26. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 For Constant Acceleration ONLY Instantaneous and average acceleration are equal. Assume time interval goes from t=0s to t At t=0s, v=v o. Can find v ave :

27. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 For Constant Acceleration ONLY Instantaneous and average acceleration are equal. Assume time interval goes from t=0s to t At t=0s, v=v o. For constant acceleration

28. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Constant Acceleration ONLY Instantaneous and average acceleration are equal. Assume time interval goes from t=0s to t At t=0s, v=v o. For constant acceleration

29. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Constant Acceleration ONLY 4 IMPORTANT EQUATIONS!!!

30. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Example A plane taking off from rest on a runway needs to achieve a speed of 35m/s in order to take off. If the acceleration of the plane is constant at 3m/s 2, what is the minimum length of the runway which can be used? How to solve: –Divide problem into “knowns” and “unknowns” KNOWN UNKNOWN

31. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Example A plane taking off from rest a runway needs to achieve a speed of 35m/s in order to take off. If the acceleration of the plane is constant at 3m/s 2, what is the minimum length of the runway which can be used? How to solve: –Divide problem into “knowns” and “unknowns” –Determine best equation to solve the problem KNOWN UNKNOWN

32. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Example A plane taking off from a runway needs to achieve a speed of 35m/s in order to take off. If the acceleration of the plane is constant at 3m/s, what is the minimum length of the runway which can be used? How to solve: –Divide problem into “knowns” and “unknowns” –Determine best equation to solve the problem –Rewrite equation KNOWN UNKNOWN

33. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Example A plane taking off from a runway needs to achieve a speed of 35m/s in order to take off. If the acceleration of the plane is constant at 3m/s, what is the minimum length of the runway which can be used? How to solve: –Divide problem into “knowns” and “unknowns” –Determine best equation to solve the problem –Rewrite equation –Input numbers KNOWN UNKNOWN

34. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Freely Falling Objects Most common example of constant acceleration is a freely falling body. The acceleration due to gravity at the Earth’s surface is basically constant and the same for ALL OBJECTS (Galileo Galilei)

35. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 Example Problem I Suppose I drop a ball off of a tower 100m high. How far has it fallen after a) 1s, b) 2s. c) How long until it hits the ground? 1) Choose coordinate system 2) Knowns and unknowns 3) Choose equation 4) Enter numbers and solve

36. Department of Physics and Applied Physics 95.141, F2010, Lecture 2 What Did We Learn Today? How to describe the position and motion of objects: –Distance travelled and Displacement –Speed and Velocity (ave. and inst.) –Acceleration (ave. and inst.) Kinematic Equations for constant a Freely Falling Objects Problem Solving