From last time…Improved experimentsUsed principle of superposition and principle of inertiaAn equationHow much longer does it take?Details of a falling objectSlow motion, in 1632How can we show this?Falling speedConstant accelerationFalling object instantaneous speed vs timeDistance vs time for falling ballGalileo’s experimentQuantifying motion: Distance and TimeThe average speedThink about this one:PowerPoint PresentationUnderstanding accelerationMajor pointsJust to check…Why a=0?What about constant acceleration?QuestionQuestionsGalileo Uniform acceleration from restFalling object: constant accelerationTough questionsDescartes’ view…Inertia and momentumDifferent types of objectsQuantifying Inertia: MomentumDescartes also said…Momentum conservationNext TimePhysics 107, Fall 2006 1From last time…Inertia: tendency of body to continue in straight-line motion at constant speed unless disturbed.Superposition: object responds independently to separate disturbances•Galileo used these properties to determine: –Light and heavy objects fall identically.–Falling time proportional to square root of falling distance.Would like to demonstrate these properties by experiment.Physics 107, Fall 2006 2Improved experiments•Penny and cotton ball experiment didn’t work because of force from the air.–Answer: Perform a better experiment that takes out the effect of the air. –In vacuum vessel or on the moon. •Falling ball experiment might also have other influences.–Height of ball when dropped.–Velocity of ball when dropped.–Slope of ramp needs to be the same.–Measuring position of ball when it lands.Should be able to improve all of these things!Physics 107, Fall 2006 3Used principle of superposition and principle of inertiaBall leaves ramp with constant horizontal speedAfter leaving ramp, it continues horizontal motion at some constant speed s (no horizontal disturbances)But gravitational disturbance causes change in vertical motion (the ball falls downward)For every second of fall, it moves to the right s metersDetermine falling time by measuring horizontal distance!Physics 107, Fall 2006 4An equationFrom this, Galileo determined that the falling time varied proportional to the square root of the falling distance. € Falling time ∝ Falling distanceFalling time = tFalling distance = d€ d ∝ t2d = ct2Physics 107, Fall 2006 5How much longer does it take?I drop two balls, one from twice the height of the other. The time it takes the higher ball to fall is how much longer than the lower ball?A. Two times longerB. Three times longerC. Four times longerD. Square root of 2 longerPhysics 107, Fall 2006 6Details of a falling object•Just how does the object fall?•Apparently independent of mass, but how fast? • Starts at rest (zero speed), ends moving fastHence speed is not constant.• 1) Falling time increases with height.• 2) Final speed increases with height.We understand how 1 works. Lets investigate 2Physics 107, Fall 2006 7Slow motion, in 1632•The inclined plane –‘Redirects’ the motion of the ball–Slows the motion down–But ‘character’ of motion remains the same. I assume that the speed acquired by the same movable object over different inclinations of the plane are equal whenever the heights of those planes are equal.Physics 107, Fall 2006 8How can we show this?•Focus on the speed at end of the ramp.•Galileo claimed this speed independent of ramp angle, as long as height is the same.Physics 107, Fall 2006 9Falling speedAs an object falls, it’s speed is A. ConstantB. Increasing proportional to timeC. Increasing proportional to time squaredPhysics 107, Fall 2006 10Constant acceleration•In fact, the speed of a falling object increases uniformly with time.•We say that the acceleration is constant•Acceleration: € Change in speedchange in time a =ΔsΔtUnits are then (meters per second)/second=(m/s)/s abbreviated m/s2Physics 107, Fall 2006 11Falling object instantaneous speed vs time•Instantaneous speed proportional to time.•So instantaneous speed increases at a constant rate •This means constant acceleration•s=at012345670 0.1 0.2 0.3 0.4 0.5 0.6 0.7TIME ( s )€ accel =change in speedchange in time=6.75 m /s0.69 s= 9.8 m /s/s = 9.8 m /s2Physics 107, Fall 2006 12Distance vs time for falling ball•Position vs time of a falling object•This completely describes the motion•Distance proportional to time squared.00.511.522.530 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Falling BallTIME ( seconds )€ d = 4.9 m /s2( )t200.511.522.530 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Falling BallTIME ( seconds )Physics 107, Fall 2006 13Galileo’s experimentA piece of wooden moulding or scantling, about 12 cubits [about 7 m] long, half a cubit [about 30 cm] wide and three finger-breadths [about 5 cm] thick, was taken; on its edge was cut a channel a little more than one finger in breadth; having made this groove very straight, smooth, and polished, and having lined it with parchment, also as smooth and polished as possible, we rolled along it a hard, smooth, and very round bronze ball.For the measurement of time, we employed a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent... the water thus collected was weighed, after each descent, on a very accurate balance; the difference and ratios of these weights gave us the differences and ratios of the times...Using this method, Galileo very precisely determined a law that explained the motionPhysics 107, Fall 2006 14Quantifying motion: Distance and Time•A moving object changes its position with time.x1 = pos. at time t1x2 = pos. at time t2x1, t1x2, t2e.g.at 10:00 am, I am 3 meters along the path (x1=3 m, t1=10:00 am)at 10:00:05 am, I am 8 meters along the path (x2=8 m, t1=10:00:05 am)My position at all times completely describes my motionPhysics 107, Fall 2006 15The average speed € Average speed = distance traveledtraveling timeAs an equation: € Distance traveled = dTraveling time = tAverage speed = s € s =dtCould also write€ d = s tSo knowing average speed lets us find distance traveledBUT maybe I walked 0 meters in the first second and then 5 meters in 4 seconds.Sometimes need instantaneous speed.Physics 107, Fall 2006 16Think about
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