physics 1710 chapter 2 motion in one dimension—ii galileo galilei linceo (1564-1642) discourses...

Physics 1710Physics 1710Chapter 2 Motion in One Dimension—II Chapter 2 Motion in One Dimension—II

Galileo Galilei LinceoGalileo Galilei Linceo

(1564-1642)(1564-1642)• Discourses and Mathematical Discourses and Mathematical Demonstrations Concerning the Demonstrations Concerning the Two New Sciences (1638) at age Two New Sciences (1638) at age

74! 74! (four years before his death at (four years before his death at

78)78)

Physics 1710Physics 1710Chapter 2 Motion in One DimensionChapter 2 Motion in One Dimension

Galileo’s RampGalileo’s Ramp


11′′ Lecture Lecture Under Under uniform accelerationuniform acceleration a axx ::

• AccelerationAcceleration is constant (hence is constant (hence “uniform”):“uniform”):

aaxx = constant = constant (>0,=0, or <0) (>0,=0, or <0)

•VelocityVelocity changes linearly in time: changes linearly in time: vvfinalfinal = v = vinitialinitial + a + ax x tt

•Displacement Displacement increases quadratically with increases quadratically with time:time:

xxfinalfinal = x = xinitialinitial+v+vinitialinitial t + ½ at + ½ axx t t 22

Displacement is the change in Displacement is the change in position.position.

ΔΔx =x = xxfinal final - x - xinitialinitial

ΔΔ is the change “operator”: is the change “operator”:

Change in Change in x =x = x at endx at end – x– x at startat start

ΔBalanceΔBalance = = BalanceBalancefinal final - - BalanceBalanceinitialinitial

REVIEW


Velocity is the Velocity is the time ratetime rate of of displacement.displacement.

Average velocity:Average velocity:vvx, ave x, ave = = ΔΔ x / x / ΔΔtt

Instantaneous velocity:Instantaneous velocity:vvx x = lim = lim ΔΔtt→∞→∞ ΔΔx /x / ΔΔt = t = dx/dtdx/dt


REVIEW

P

osit

ion

(m

)

Time (sec)

Δx

Δt vaverage =Δx/Δt


VelocityVelocityP

osit

ion

(m

)

Time (sec)

vvx, ave x, ave = = ΔΔx / x / ΔΔtt

AverageAverageVelocityVelocity

vvx x = = dx/dtdx/dt

Instantaneous Instantaneous VelocityVelocity

ΔΔtt

ΔΔx x

Plot it!


Acceleration is the Acceleration is the time ratetime rate of of changechange of of velocity.velocity.

Average acceleration:Average acceleration:aax, ave x, ave = = ΔΔvvxx / / ΔΔt t

= (v= (vx,final x,final -v-vx, initial x, initial )/ )/ ΔΔt t

Instantaneous acceleration:Instantaneous acceleration:aax x = lim = lim ΔΔtt→∞→∞ ΔΔvvxx / / ΔΔt t

= dv= dvx x /dt/dt

aax x = dv= dvx x /dt /dt =d(dx=d(dxx x /dt)/dt /dt)/dt = d= d22x/dtx/dt 2 2


Motion MapMotion MapFrom “snap shots” of motion at equal From “snap shots” of motion at equal

intervals of time we can determine the intervals of time we can determine the displacement, the average velocity and the displacement, the average velocity and the

average acceleration in each case.average acceleration in each case.

Uniform motion

Accelerated motion


VelocityVelocity

Physics 1710Physics 1710Chapter 2 Motion in One DimensionChapter 2 Motion in One Dimension

Posit

ion

(m

)

Time (sec)

Plot them!

Velo

cit

y

(m/s

ec)

Time (sec)

a1= 0

a2>0

vvx x = = dx/dtdx/dt

aax x = = dvdvx x /dt/dt


Galileo Galilei Galileo Galilei LinceoLinceo

Galileo’s rule of odd numbers:Galileo’s rule of odd numbers:

Under Under uniform accelerationuniform acceleration from from restrest, ,

a body will a body will traverse distancestraverse distances in in successive successive equalequal intervals of timeintervals of time

that stand in that stand in ratioratio

as the as the odd numbersodd numbers 1,3,5,7,9 … 1,3,5,7,9 …


Galileo’s Ramp Demonstration


0 : 0 : 0 = 0 0 = 0 22

11 : : 0+1 =1 = 1 0+1 =1 = 1 22

22 : : 1+3 = 4 = 2 1+3 = 4 = 2 22

33 : : 4+5 = 9 = 3 4+5 = 9 = 3 22

44 : : 9+7= 16 = 4 9+7= 16 = 4 22

55 :: 16+9= 25 = 5 16+9= 25 = 5 22

66 :: 25+11= 36 = 6 25+11= 36 = 6 22

ObservationObservation::

∆∆x x ∝ ∝ t t 22 ; from rest (v ; from rest (vinitialinitial = 0) = 0)



Kinematic Equations from Calculus:Kinematic Equations from Calculus:(dv/dt) = a = constant(dv/dt) = a = constant

∫∫00tt (dv/dt) dt = ∫ (dv/dt) dt = ∫00

tt a dt a dt

∫∫vvinitialinitial

vvfinalfinal dv = a (t-0) dv = a (t-0)

∆∆v = v– vv = v– vinitialinitial = at = at

The The change ichange in the n the instantaneous velocityinstantaneous velocity is is

equal to the (constant) equal to the (constant) accelerationacceleration multiplied multiplied

by its by its durationduration..

v= vv= vinitialinitial + at + at


Kinematic Equations from Calculus:Kinematic Equations from Calculus:dx/dt = v dx/dt = v

∫∫00tt (dx/dt) dt = ∫ (dx/dt) dt = ∫00

tt v v dtdt

xx – x– xinitial initial = ∫= ∫00tt (v (vinitial initial + at) dt+ at) dt

= = vvinitial initial (t-0) + ½ a(t (t-0) + ½ a(t 22-0)-0)

∆∆x = vx = vinitial initial t + ½ at t + ½ at 22

The displacement under uniform acceleration is The displacement under uniform acceleration is equal to the displacement at constant velocity equal to the displacement at constant velocity plus plus one halfone half the the accelerationacceleration multiplied by the multiplied by the

square of its durationsquare of its duration..

xx == x xinitial initial + v+ vinitial initial t + ½ at t + ½ at 22


Kinematic Equations from Calculus:Kinematic Equations from Calculus:v v 22 = (v = (vinitialinitial + at) + at) 2 2

v v 22 = v = vinitial initial 2 2 +2 v +2 vinitialinitial at + a at + a 22 t t 22

v v 22 = v = vinitial initial 22 + + 2a2a (v (vinitial initial t + ½ at t + ½ at 22))

v v 22 = v = vinitial initial 22 +2a∆x +2a∆x

The change in the square of the velocity is equal The change in the square of the velocity is equal to two times the acceleration multiplied by the to two times the acceleration multiplied by the distance over which the acceleration is applied.distance over which the acceleration is applied.

Kinematic Equations (1-D Uniform a):Kinematic Equations (1-D Uniform a):


vvx finalx final = v = vx initialx initial + a + ax x tt

x x finalfinal = x = x initialinitial + v + vx initialx initial t + ½ a t + ½ axxt t 22

vvx finalx final 2 2 = v= vx initialx initial 2 2 + 2 a+ 2 axx (x (x finalfinal - x - x

initialinitial ) )

80/20 facts:80/20 facts:


In In free fallfree fall near the Earth, all near the Earth, all bodies are accelerated uniformly bodies are accelerated uniformly downward with an acceleration of downward with an acceleration of

aazz = - g = -9.80 m/s = - g = -9.80 m/s22..

80/20 facts:80/20 facts:

Plot it!


Posit

ion

(m

)

Velo

cit

y

(m/s

ec)

Time (sec)

Accele

rati

on

(m

/s/s

)00

dx/dt dx/dt →→

dv/dt dv/dt →→

00

-9.8 m/s/s-9.8 m/s/s

Time (sec)

Time (sec)

• The change in the instantaneous velocity is equal to The change in the instantaneous velocity is equal to the (constant) acceleration multiplied by its duration. the (constant) acceleration multiplied by its duration. ∆v = at∆v = at

•The displacement is equal to the displacement at The displacement is equal to the displacement at constant velocity plus one half of the product of the constant velocity plus one half of the product of the acceleration and the square of its duration. acceleration and the square of its duration. ∆x = ∆x = vvinitialinitial t + ½ at t + ½ at 22

•The change in the square of the velocity is equal to The change in the square of the velocity is equal to two times the acceleration multiplied by the distance two times the acceleration multiplied by the distance traveled during acceleration. traveled during acceleration. ∆v ∆v 22 = 2a ∆x = 2a ∆x

•The acceleration of falling bodies is 9.8 m/s/s The acceleration of falling bodies is 9.8 m/s/s downward.downward.

a = - g = - 9.8 m/s/sa = - g = - 9.8 m/s/s

Summary:Summary:


• The main point of today’s lecture.The main point of today’s lecture.

• A realization I had today.A realization I had today.

•A question I have. A question I have.

11′ Essay′ Essay::

One of the following:One of the following:


physics 1710 chapter 2 motion in one dimension—ii galileo galilei linceo (1564-1642) discourses...

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