1. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Physics I 95.141 LECTURE 4 9/15/10

2. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Exam Prep Problem Vince Carter’s vertical leap is 43”. –A) (10pts) With what initial vertical velocity does Carter leave the ground? –B) (10pts) What is his hang time? –C) (10pts) Assuming Carter leaps straight up at t=0s and lands at just after t=T, draw the vectors for James’ displacement, velocity, and acceleration at: i) The instant he leaves the ground (t=0s) ii) t=T/4 iii) t=T/2 iv) t=3T/4 v) t=T

3. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Exam Prep Problem Vince Carter’s vertical leap is 43”. –A) (10pts) With what initial vertical velocity does James leave the ground?

4. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Exam Prep Problem Vince Carter’s vertical leap is 43”. –B) (10pts) What is his hang time?

5. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Exam Prep Problem Vince Carter’s vertical leap is 43”. –C) (10pts) Assuming Carter leaps straight up at t=0s and lands at just after t=T, draw the vectors for James’ displacement, velocity, and acceleration at:

6. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Outline Lecture 3 Review Vector Kinematics Relative Motion What do we know? –Units/Dimensions/Measurement/SigFigs –Kinematic equations –Freely falling objects –Vectors

7. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Lecture 3 Review Freely falling body problems –Batman’s bat-hook Scalars and Vectors Graphical description of vectors and vector addition. Vector components Mathematical description of vector addition (addition of components) Unit Vectors

8. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Vector Kinematics We can now do kinematics in more than one dimension –This is helpful, because we live in a 3D world! We previously described displacement as Δx, but this was for 1D, where motion could only be positive or negative. In more than 1 dimension, displacement is a vector

9. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Vector Kinematics Now, instead of describing displacement in terms of either vertical or horizontal position, we can talk about a displacement vector!

10. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Vector Kinematics In unit vectors, we can write the displacement vector as: We can now rewrite our expression for average velocity:

11. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Vector Kinematics Average velocity only tells part of the story Just like for motion in 1D, we can let Δt get smaller and smaller…. Gives instantaneous velocity vector:

12. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Velocity Vector The magnitude of the average velocity vector is NOT equal to the average speed. But the magnitude of the instantaneous velocity vector is equal to the instantaneous speed at that time

13. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Instantaneous Velocity (math) To find the instantaneous velocity, we can take the derivative of the position vector with respect to time:

14. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Example Say we are given the position of an object to be: Can we find the velocity as a function of time?

15. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Acceleration Vector Average acceleration: Instantaneous acceleration

16. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Example Problem Imagine we are given the position of an object as a function of time –Find displacement at t=1s and t=3s –Find velocity and acceleration as a function of time –Find velocity and acceleration at t=3s

17. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Example Problem Imagine we are given the position of an object as a function of time –Find displacement at t=1s and t=3s –Find velocity and acceleration as a function of time –Find velocity and acceleration at t=3s

18. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Example Problem Imagine we are given the position of an object as a function of time –Find displacement at t=1s and t=3s –Find velocity and acceleration as a function of time –Find velocity and acceleration at t=3s

19. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Example Problem Let’s say we are told that a Force causes an object to accelerate in the -y direction at 5m/s 2. The object has an initial velocity in the +x direction of 10m/s, and in the +y direction of 15 m/s, and starts at the point (0,0). –A) Give the initial velocity vector of the object –B) Plot x(t) vs. t –C) Plot y(t) vs. t –D) Plot the object’s trajectory in the xy plane

20. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Example Problem Let’s say we are told that a Force causes an object to accelerate in the -y direction at 5m/s 2. The object has an initial velocity in the +x direction of 10m/s, and in the +y direction of 15 m/s, and starts at the point (0,0).. –A) Give the initial velocity vector of the object 20 vxvx vyvy

21. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Example Problem Let’s say we are told that a Force causes an object to accelerate in the -y direction at 5m/s 2. The object has an initial velocity in the +x direction of 10m/s, and in the +y direction of 15 m/s, and starts at the point (0,0). –Before we solve B-D, let’s determine equations of motion (METHOD I)

22. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Example Problem Let’s say we are told that a Force causes an object to accelerate in the -y direction at 5m/s 2. The object has an initial velocity in the +x direction of 10m/s, and in the +y direction of 15 m/s, and starts at the point (0,0). –Before we solve B-D, let’s determine equations of motion (METHOD II)

23. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Example Problem Let’s say we are told that a Force causes an object to accelerate in the -y direction at 5m/s 2. The object has an initial velocity in the +x direction of 10m/s, and in the +y direction of 15 m/s, and starts at the point (0,0). –B) Plot x(t) vs t 10 100 t(s) x(t) timex(t) 00m 110m 220m 550m 10100m

24. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Example Problem Let’s say we are told that a Force causes an object to accelerate in the -y direction at 5m/s 2. The object has an initial velocity in the +x direction of 10m/s, and in the +y direction of 15 m/s, starts at (0,0). –C) Plot y(t) vs t timey(t) 00m 112.5m 220m 322.5m 420m 512.5 10-100

25. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Example Problem D) Plot object trajectory Choose coordinate system timex(t)y(t) 00m 110m12.5m 220m 330m22.5m 440m20m 55012.5 10100-100

26. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Relative Velocity So far we have looked at adding displacement vectors May also find situations where we need to add velocity vectors

27. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Relative Velocity Two velocities: 5m/s 25m/s

28. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Relative Velocity In this case, our hero would presumably prefer not to be decapitated by the bridge So we are interested in his velocity relative to the bridge He is on a train moving at +25 m/s relative to the bridge His velocity relative to the train is -5m/s So his velocity relative to the bridge is:

29. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Relative Velocity What is Kirk’s velocity when he hits the ground? –Assume he leaps when car is moving 20m/s

30. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Relative Velocity (Example 1) Imagine you are on a barge floating down the river with the current You walk diagonally across the barge with a velocity What is your velocity with respect to the water? With respect to the river bank?

31. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Another River Problem A boat’s speed in still water is 1.85m/s. If you want to directly cross a stream with a current 1.2m/s, what upstream angle should you take?

32. Department of Physics and Applied Physics 95.141, F2010, Lecture 4 Today We Learned…. Vector kinematics –Displacement vector –Average velocity vector –Inst. Velocity vector –Average acceleration vector –Inst. Acceleration vector –Vector equations of motion Relative Velocity