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