Kinematics in Two Dimensions

1. Kinematics in Two Dimensions • Position, velocity, acceleration vectors • Constant acceleration in 2-D • Free fall in 2-D Serway and Jewett : 4.1 to 4.3 Physics 1D03 - Lecture 4

2. The components of are the coordinates (x,y) of the particle: For a moving particle, , x(t), y(t) are functions of time. y The Position vectorpoints from the origin to the particle. path (x,y) yj xi x Physics 1D03 - Lecture 4

3. y Displacement : final vavg initial Average Velocity : (a vector parallel to ) x y Instantaneous Velocity : is tangent to the path of the particle v x Physics 1D03 - Lecture 4

4. path of particle Acceleration is the rate of change of velocity : Physics 1D03 - Lecture 4

5. a is the rate of change of v (Recall: a derivative gives the “rate of change” of function wrt a variable, like time). Velocity changes if i) speed changes ii) direction changes (even at constant speed) iii) both speed and direction change In general, acceleration is not parallel to the velocity. Physics 1D03 - Lecture 4

6. At which positions is ?(consider tangential a only!) 1 5 4 2 3 Concept Quiz A pendulum is released at (1) and swings across to (5). a) at 3 only b) at 1 and 5 only c) at 1, 3, and 5 d) none of the above Physics 1D03 - Lecture 4

7. Components: Each vector relation implies 3 separate relations for the 3 Cartesian components. (i, j, k, are unit vectors) We get velocity components by differentiation: the unit vectors are constants Physics 1D03 - Lecture 4

8. Each component of the velocity vector looks like the 1-D “velocity” we saw earlier. Similarly for acceleration: Physics 1D03 - Lecture 4

9. Common Notation – for time derivatives only, a dot is often used: Physics 1D03 - Lecture 4

10. If is constant (magnitude and direction), then: Where are the initial values at t = 0. Constant Acceleration + Projectile Motion In 2-D, each vector equation is equivalent to a pair of component equations: Example: Physics 1D03 - Lecture 4

11. Shooting the Gorilla Tarzan has a new AK-47. George the gorilla hangs from a tree branch, and bets that Tarzan can’t hit him. Tarzan aims at George, and as soon as he shoots his gun George lets go of the branch and begins to fall. Where should Tarzan be aiming his gun as he fires it?A) above the gorillaB) at the gorillaC) below the gorilla Physics 1D03 - Lecture 4

12. a=g v0t (1/2)gt2 v0 r0 r(t) =r0+v0t +(1/2)gt2 Physics 1D03 - Lecture 4

13. 2 seconds after firing • 100 seconds after firing • seconds after firing • Other (explain) 100 m/s 2 m/s Concept quiz Your summer job at an historical site includes firing a cannon to amuse tourists. Unfortunately, the cannon isn’t properly attached, and as the cannonball shoots forward (horizontally) the cannon slides backwards off the wall. If the cannon hits the ground 2 seconds later, the cannonball will hit the ground: Physics 1D03 - Lecture 4

14. Example Problem A stone is thrown upwards from the top of a 45.0 m high building with a 30º angle above the horizontal. If the initial velocity of the stone is 20.0 m/s, how long is the stone in the air, and how far from the base of the building does it land ? Physics 1D03 - Lecture 4

15. 100 m/s 30° d 20° Example Problem: Cannon on a slope. How long is the cannonball in the air, and how far from the cannon does it hit? Try to do it two different ways: once using horizontal and vertical axes, once using axes tilted at 20o. Physics 1D03 - Lecture 4

16. Show that for: d vo θ Φ Physics 1D03 - Lecture 4

17. Summary • position vector points from origin to a particle • velocity vector • acceleration vector • for constant acceleration, we can apply 1-D formulae to each component separately • for free fall in uniform , horizontal and vertical motions are independent Physics 1D03 - Lecture 4