VIEWS: 54 PAGES: 17 POSTED ON: 4/24/2011 Public Domain
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 1 y path The Position vector r points from the origin to the particle. (x,y) yj r xi x The components of r are the coordinates (x,y) of the particle: r x i y j For a moving particle, r (t ), x(t), y(t) are functions of time. Physics 1D03 - Lecture 4 2 Displacement : r rf ri y final vavg r Average Velocity : rf initial v avg r / t ri (a vector parallel to r ) x Instantaneous Velocity : y v dr / dt is tangent to the v path of the particle r x Physics 1D03 - Lecture 4 3 Acceleration is the v (t t ) rate of change of v (t ) velocity : time t t dv time t a dt path of particle v lim v t 0 t a t v v (t ) v (t t ) Physics 1D03 - Lecture 4 4 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 5 Concept Quiz A pendulum is released at (1) and swings across to (5). At which positions is a 0 ? (consider tangential a only!) a) at 3 only b) at 1 and 5 only c) at 1, 3, and 5 1 5 d) none of the above 2 3 4 Physics 1D03 - Lecture 4 6 Components: Each vector relation implies 3 separate relations for the 3 Cartesian components. r x i y jzk (i, j, k, are unit vectors) We get velocity components by differentiation: dr v dt dx i dy j dz k the unit vectors are constants dt dt dt v xi v y j v z k Physics 1D03 - Lecture 4 7 Each component of the velocity vector looks like the 1-D “velocity” we saw earlier. Similarly for acceleration: dv dvx dv y dvz a i j k dt dt dt dt dx dvx d 2 x vx , ax 2 dt dt dt dy dv y d 2 y vy , ay 2 dt dt dt dz dvz d 2 z vz , az 2 dt dt dt Physics 1D03 - Lecture 4 8 Common Notation – for time derivatives only, a dot is often used: dr v r dt dv r a v dt Physics 1D03 - Lecture 4 9 Constant Acceleration + Projectile Motion If a is constant (magnitude and direction), then: v (t ) v o a t r (t ) ro v o t 2 1 a t2 Where ro , v o are the initial values at t = 0. In 2-D, each vector equation is equivalent to a pair of component equations: x(t ) xo vox t 12 a x t 2 y (t ) yo voy t 12 a y t 2 Example: Free fall : a g 9.8 m/s [down] 2 Physics 1D03 - Lecture 4 10 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 gorilla B) at the gorilla C) below the gorilla Physics 1D03 - Lecture 4 11 a=g v0t v0 (1/2)gt2 r0 r(t) =r0+v0t +(1/2)gt2 Physics 1D03 - Lecture 4 12 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: a) 2 seconds after firing 2 m/s 100 m/s b) 100 seconds after firing c) 2 100 seconds after firing d) Other (explain) Physics 1D03 - Lecture 4 13 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 14 Example Problem: Cannon on a slope. 100 m/s 30° 20° 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 15 Show that for: d vo θ Φ 2v cos sin( ) 2 d o g cos 2 Physics 1D03 - Lecture 4 16 Summary • position vector r points from origin to a particle dr • velocity vector v dt dv v • acceleration vector a [ , as t , v go to zero] dt t • for constant acceleration, we can apply 1-D formulae to each component separately • for free fall in uniform g , horizontal and vertical motions are independent Physics 1D03 - Lecture 4 17