2D Kinematics. Note not covering scalar product or vector product right now we will need it for material in Chap 7 and it will be covered then.

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1 Announcements: 2D Kinematics CAPA due at 10pm tonight There will be the third CAPA assignment ready this evening. Chapter 3 on Vectors Note not covering scalar product or vector product right now we will need it for material in Chap 7 and it will be covered then. Web page: 1

2 Clicker question 1 Set frequency to BA Two stones are dropped into a bottomless pit, the second stone is dropped 2 seconds after the first stone. Assume no air resistance. As both stones fall, the difference in their velocities.. A: increases B: decreases C: remains constant A tough one! Both balls are dropped from y 0 =0, with v 0 =0, so we have v1= -g*t v2 = -g*(t-2). So the difference in velocities is just 2*g, a constant. That is, after 2 seconds, ball one is falling at -g*2 sec = -20 m/s, and from then on, they each gain the same amount of extra velocity (9.8 m/s) each second. They both speed up, but always the leading one is exactly 202 m/s faster.

3 Clicker question 2 Set frequency to BA As both stones fall, the difference in their heights (ypositions).. A: increases B: decreases C: remains constant Since the lower one is ALWAYS faster, it is constantly gaining on the other one! The difference in their heights increases! 3

4 Clicker question 3 Set frequency to BA Q. The solid line has length A and makes an angle θ with the negative y-axis. What is the length of the dotted line? A. A cosθ B. A sinθ C. A tanθ Hypotenuse D. sinθ/a E. cosθ/a Opposite A θ y Adjacent x Adjacent 4

5 Vector addition by components Split vectors into orthogonal components and add components individually. For and then 5

6 Vector addition by components (2) Use trigonometry to split vectors into orthogonal components if you are given magnitude and direction Draw quick diagram and determine angle but angles can be tricky Need to determine how measured angle relates to angle from +x axis. In this case C = (7.2,154 ) 6

7 Position and velocity vectors Position vector gives the distance and direction from the origin to the particle position y(cm) Can write in terms of components: x(cm) or 7

8 Velocity vectors Velocity is the change in position over change in time Average velocity: Instantaneous velocity: Often, motion along one axis is independent of motion along other axes (with proper coordinate choice) which allows us to separate the vector equation into components: 8

9 Position and velocity vectors A ball is moving across a level surface and its position vector is recorded at times 1,2,3,4 seconds. How can we find the average velocity between times 1 and 2? Average velocity is and so we need. Can calculate from vector addition. Same as vector connecting the two vector end points y(cm) x(cm) 9

10 Position and velocity vectors Note the average velocity is the same between any of the points. Appears the velocity is constant. Can measure to determine the average velocity y(cm) 5 5 x(cm) In this case we could redefine the coordinate system to get a 1D scenario with constant velocity 10

11 Acceleration vectors Velocity deals with the change of the position vector and acceleration deals with the change of the velocity vector. Average acceleration: Instantaneous acceleration: Again, with a proper coordinate choice the components separate so we end with the simplified result: 11

12 Linear acceleration: velocity magnitude changes but direction stays the same. Gravity is an example. y(cm) Acceleration vectors Centripetal acceleration: velocity magnitude constant but direction changes. Example: car rounding a corner at constant speed. y(cm) x(cm) x(cm) Generally, acceleration will be a combination of both 12

13 Clicker question 4 Set frequency to BA Q. A particle is moving at constant speed along the path shown. Its velocity vector at two different times is shown. What is X the direction of the acceleration when the particle is at point X? B it points in the same A D C (E) None of these direction as 13

14 Projectile motion This describes the motion of a body (bullet, basketball, motorcycle, etc.) in free fall after being launched. The only acceleration is due to gravity and is always straight down. Thus, the velocity in the horizontal direction is constant In the vertical direction there is acceleration from gravity 14

15 Clicker question 5 Set frequency to BA Q. A basketball is launched from a basketball cannon and follows the trajectory shown. What is the direction of the acceleration at point X? A E B D C X Once launched, the only acceleration is due to gravity and is straight down. The horizontal velocity is constant 15

16 Velocities in projectile motion Initial velocity in horizontal and vertical directions depends on angle α and speed v 0 of launch. and α Solving projectile motion problems Realize the horizontal and vertical motions are independent. Their only connection is through the time the projectile is in the air. 16

17 Solving a projectile motion problem A basketball launched on a level surface travels 15 m and reaches a maximum height of 6.4 m. What α is the initial velocity? 1. Draw a diagram 2. Figure out what we know 3. Figure out what we need: To get v 0x we need the flight time to use Can get v 0y from with y=y max and v y =0 Can now use to get t 17