1. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Physics I 95.141 LECTURE 2 9/9/09

2. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Administrative Notes Section 206 (TR 11:30 a.m) has been moved from Ball 313 to Olsen 408 Please make sure you enter your UML ID in your PRS without leading 0’s –If your ID is 00123456 –Enter 123456! LP Homeworks due before each lecture. Instruction sheet at class website. HWs due in recitation. 1 st HW due tomorrow (or today after class if you have MW recitation).

3. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Outline Reference Frames and Displacement Average velocity Instantaneous velocity Acceleration Motion with constant acceleration Freely falling bodies

4. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Frames of Reference 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

5. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Frames of Reference Frames of reference not just important for position, but for speed and velocity as well! 5m/s 25m/s

6. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Frames of Reference 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, F2009, Lecture 2 Distance vs. Displacement Distance: the total distance traveled by an object along its path Displacement: how far an object is from its starting point

8. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Displacement Displacement is a vector. A vector is a quantity that has both magnitude and direction.

9. Department of Physics and Applied Physics 95.141, F2009, 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

10. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Average Velocity Velocity has both a magnitude and a direction. It is a vector. Suppose this walk took 50 seconds

11. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Average Velocity Average velocity is given by the displacement divided by the elapsed time: Example: A runner takes 30 seconds to complete the loop. What is the runner’s average speed and velocity? R=3m

12. Department of Physics and Applied Physics 95.141, F2009, 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. Say you took 30 minutes. This means you didn’t speed (T/F)

13. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Instantaneous vs. Average Velocity Mathematically, the instantaneous velocity is the derivative of the position function Graphically, speed is the slope of the x vs t plot. A derivative gives you the slope at a point t.

14. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Instantaneous Velocity 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

15. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Example Problem a)Determine displacement between t 1 =3s and t 2 =5s b)Determine average velocity during this time c)Determine magnitude of instantaneous velocity at t 2 =5s.

16. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Example Problem a)Determine displacement between t 1 =3s 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, F2009, Lecture 2 Acceleration Even knowing the velocity doesn’t tell us the entire story! Velocity can change with time: this is called acceleration! Average Acceleration:

18. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Example Problem Car accelerates from rest to 90km/hr in 5s. What is average acceleration? Units! 5m/s 2

19. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 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

20. Department of Physics and Applied Physics 95.141, F2009, 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.

21. Department of Physics and Applied Physics 95.141, F2009, 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)). x(t)=2.1t 2 +2.8 a)What is average acceleration from t 1 =3 to t 2 =5s? b)What is a(t)? 4.2 m/s 2

22. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Constant Acceleration Instantaneous and average acceleration are equal. Assume time interval goes from t=0s to t. For constant acceleration

23. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Constant Acceleration Instantaneous and average acceleration are equal. Assume time interval goes from t=0s to t. For constant acceleration

24. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Constant Acceleration 4 IMPORTANT EQUATIONS!!!

25. Department of Physics and Applied Physics 95.141, F2009, 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” KNOWN UNKNOWN

26. Department of Physics and Applied Physics 95.141, F2009, 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 KNOWN UNKNOWN

27. Department of Physics and Applied Physics 95.141, F2009, 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

28. Department of Physics and Applied Physics 95.141, F2009, 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

29. Department of Physics and Applied Physics 95.141, F2009, 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) If I drop a ball and piece of paper at the same time, which hits the ground first? –A) Ball –B) Paper –C) Same time

30. Department of Physics and Applied Physics 95.141, F2009, 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) OK, let’s try once more, but I will crumple up the paper. Which lands first? –A) Ball –B) Paper –C) Same time

31. Department of Physics and Applied Physics 95.141, F2009, 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 a)4.9m b) 19.6m c) 4.52s

32. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Example Problem II You throw a ball up with an initial velocity of 10m/s. A) How long does it take to get to 5m above the release point? B) How long until it reaches the top of its flight? C) How long until it returns to your hand? 1) Choose coordinate system 2) Knowns and unknowns 3) Choose equation(s)

33. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Example Problem II a)0.876s and 1.165s b) 1.02s c) 0s and 2.04s 4) Enter numbers and solve

34. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Sign of Acceleration and Velocity If velocity is positive, acceleration can be positive or negative For instance, braking car Or, ball thrown up in the air