Chapter 4, Motion in 2 Dimensions

1. Chapter 4, Motion in 2 Dimensions

2. Position, Velocity, Acceleration • Just as in 1d, in 2, object’s motion is completely known if it’s position, velocity, & acceleration are known. • Position Vector r • In terms of unit vectors discussed last time, for object at position (x,y) in x-y plane: r  x i + y j Object moving: r depends on time t: r = r(t) = x(t) i + y(t) j

3. Object moves from A (ri) to B (rf) in x-y plane: • Displacement Vector  Δr = rf - ri If this happens in time Δt = tf - ti • Average Velocity vavg  (Δr/Δt) Obviously, in the same direction as displacement. Independent of path between A & B

4. As Δt gets smaller & smaller, clearly, A & B get closer & closer together. Just as in 1d, we define the instantaneous velocity as:  velocity at any instant of time  average velocity over an infinitesimally short time • Mathematically, instantaneous velocity: v = lim∆t  0[(∆r)/(∆t)] ≡ (dr/dt) lim ∆t  0  ratio (∆r)/(∆t) for smaller & smaller ∆t. Mathematicians call this a derivative.  Instantaneous velocity v≡time derivative of displacementr

5. Instantaneous velocityv=(dr/dt). • Magnitude |v| of vector v ≡ speed. As motion progresses, speed & direction of v can both change. Object moves from A (vi) to B (vf) in x-y plane: Velocity Change Δv = vf - vi This happens in timeΔt = tf - ti • Average Acceleration aavg  (Δv/Δt) As both speed & direction of v change, arbitrary path

6. As Δt gets smaller & smaller, clearly, A & B get closer & closer together. Just as in 1d, we define instantaneous acceleration as:  acceleration at any instant of time  average acceleration over infinitesimally short time • Mathematically, instantaneous acceleration: a = lim∆t  0[(∆v)/(∆t)] ≡ (dv/dt) lim ∆t  0  ratio (∆v)/(∆t) for smaller & smaller ∆t. Mathematicians call this a derivative.  Instantaneous acceleration a≡time derivative of velocityv

7. 2d Motion, Constant Acceleration • Can show: Motion in the x-y plane can be treated as 2 independent motions in the x & y directions.  Motion in the x direction doesn’t affect the y motion & motion in the y direction doesn’t affect the x motion.

8. Object moves from A (ri,vi), to B (rf,vf), in x-y plane. Position changes with time: Acceleration a is constant, so, as in 1d, can write (vectors!):  rf = ri + vit + (½)at2 Velocity changes with time: Acceleration a is constant, so, as in 1d, can write (vectors!):  vf = vi + at

9. Acceleration a is constant, (vectors!):  rf = ri + vit + (½)at2, vf = vi + at Horizontal Motion: xf = xi + vxit + (½)axt2,vxf = vxi + axt Vertical Motion: yf = yyi + vyit + (½)ayt2,vyf = vyi + ayt

10. Projectile Motion

11. Equations to Use • One dimensional, constant acceleration equations for x & y separately! • x part: Acceleration ax = 0! • y part: Acceleration ay = g(if take down as positive). • Initial x & y components of velocity: vxi & vyi. x motion: vxf = vxi = constant. xf = xi + vxi t y motion: vyf = vyf + gt, yf = yi + vyi t + (½)g t2 (vyf) 2 = (vyi)2 + 2g (yf - y0)

12. Projectile Motion • Simplest example:Ball rolls across table, to the edge & falls off edge to floor. Leaves table at time t = 0. Analyze y part of motion & x part of motion separately. • y part of motion: Down is positive & origin is at table top: yi = 0. Initially, no y component of velocity: vyi = 0  vyf = gt, yf = (½)g t2 • x part of motion: Origin is at table top: xf = 0. No x component of acceleration(!): ax = 0. Initially x component of velocity is: vxf  vxf = vxi , xf = vxit

13. Ball Rolls Across Table & Falls Off

14. Summary:Ball rolling across the table & falling. • Vector velocity v has 2 components: vxf = vxi , vyf = gt • Vector displacement D has 2 components: xf = vxft , yf = (½)g t2

15. Projectile Motion • PHYSICS:y part of motion: vyf = gt , yf = (½)g t2 SAME as free fall motion!!  An object projected horizontally will reach the ground at the same time as an object dropped vertically from the same point! (x & y motions are independent)

16. Projectile Motion • General Case:

17. General Case: Take y positive upward& origin at the point where it is shot: xi = yi = 0 vxi = vicosθi, vyi = visinθi • Horizontal motion: NO ACCELERATION IN THE x DIRECTION! vxf = vxi , xf = vxi t • Vertical motion: vyf = vyi - gt , yf = vyi t - (½)g t2 (vyf) 2 = (vyi)2 - 2gyf • If y is positive downward, the - signs become + signs. ax = 0, ay = -g = -9.8 m/s2