Kinematics in Two Dimensions

1. Kinematics in Two Dimensions AP Physics B Chapter 3 Notes

2. What is projectile motion? Projectile Motion-The path of any object which is projected by some means and continues to move due to its own inertia (mass). • Motion in more than one direction • Near the surface of the earth

3. First need to discuss vectors… • Vectors have directionand magnitude (size) • Examples: • Scalar quantities have no direction associated with them • Examples:

4. Addition of vectors--Graphical Method • For vectors in the same dimension, simple addition and subtraction works • Pay attention to sign (direction)

5. Addition of vectors--Graphical Method • For vectors in the two dimensions, it is a little more complicated • If the vectors are at right angles we can use Pythagorean’s theorem—order of addition does not matter

6. Addition of vectors--Graphical Method • For vectors not at right angles, use the “tip to tail” method

7. Vector Math • Addition of vectors follows most usual arithmetic rules: A+B = B+A(A+B)+C = A+(B+C)A-B = A+(-B)

8. Vector Math • A vector V can be multiplied by a scalar c; the result is a vector cV that has the same direction but a magnitude cV. If c is negative, the resultant vector points in the opposite direction.

9. Adding vectors—Component Method • Any vector can be expressed as the sum of components vectors--the other vectors are chosen so that they are perpendicular to each other (x and y axes).

10. Adding vectors—Component Method • Use of simple trig functions allows you to calculate the x- and y-components of any vector

11. Adding vectors—Example • Problem 10 pg. 66: Find the resultant vector in terms of a) components and b) magnitude and angle with the x axis.

12. Relative Velocity • A boat trying to cross a river has a speed of vBWrelative to the water • The river has a speed of vWSrelative to the shore • The boat’s resultant velocity relative to the shore is

13. Relative Velocity--Example • Problem 37 p.68: Huck Finn walks 0.6 m/s across his raft which is traveling down the river at 1.7 m/s relative to the river bank. What is Huck’s velocity relative to the river bank?

14. Projectile motion—horizontal launch • Use of simple trig functions allows you to calculate the x- and y-components of any vector • Projectile motion can be broken into horizontal (x) and vertical (y) components • The two component vectors are independent of each other

15. Horizontal “Velocity” Component • NEVER changes--initial horizontal velocity equals the final horizontal velocity In other words, the horizontal velocity is CONSTANT. BUT WHY? Gravity DOES NOT work horizontally to increase or decrease the velocity.

16. Vertical “Velocity” Component • Changes (due to gravity), does NOT cover equal displacements in equal time periods. As a projectile descends, vertical velocity increases due to g

17. Projectile motion—horizontal launch • Speed in the x-direction is constant • Motion in the y-direction has constant acceleration of g • The two ball were released at the same time and have same y-positions over time

18. Horizontal launch--Example • Problem 63 pg. 70: A cliff diver pushes off horizontally from the top of a 35 m cliff and must clear rocks 5 m out from the base. What minimum push off speed is required and how long are they in the air?

19. Projectile motion—angled launch • If an object is launched at an angle θ to the horizontal, the analysis is similar except that there is an initial vertical velocity

20. Angle launch—Example • Problem 30 pg. 67: A projectile is fired with an initial speed of 65.2 m/s at an angle of 34.5º above horizontal. Determine a) maximum height, b) total time in the air, c) its range, d) its velocity 1.5 s after firing.

21. Projectile motion—key points • Projectile motion is a parabola with symmetry about the top of the arc • vyis zero at the top • vx is constant • Time to the peak is one-half total time (for same dy )

22. Projectile motion—summary equations • The same kinematic equations from Ch. 2 apply