chapter 4, motion in 2 dimensions. position, velocity, acceleration just as in 1d, in 2,...
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Chapter 4, Motion in 2 Dimensions
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
• 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
• 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 displacement r
• Instantaneous velocity v = (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 time Δt = tf - ti
• Average Acceleration aavg (Δv/Δt) As both speed & direction of v change, arbitrary path
• 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 velocity v
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.
• 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
• 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
Projectile Motion
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 ty motion: vyf = vyf + gt, yf = yi + vyi t + (½)g t2
(vyf) 2 = (vyi)2 + 2g (yf - y0)
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
Ball Rolls Across Table & Falls Off
• 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
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)
Projectile Motion• General Case:
• 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
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