1Physics 107, Fall 2006 1From last time…Inertia:tendency of body to continue in straight-line motion at constantspeed 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 theseproperties by experiment.Physics 107, Fall 2006 2Improved experiments• Penny and cotton ball experiment didn’twork because of force from the air.– Answer: Perform a better experiment that takesout the effect of the air.– In vacuum vessel or on the moon.• Falling ball experiment might also haveother 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 andprinciple of inertiaBall leaves ramp withconstant horizontal speedAfter leaving ramp, it continues horizontal motion at someconstant speed s (no horizontal disturbances)But gravitational disturbance causes change in verticalmotion (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 timevaried proportional tothe square root of the falling distance. Falling time Falling distanceFalling time = tFalling distance = dd t2d = ct2Physics 107, Fall 2006 5How much longer does it take?I drop two balls, one from twice the height ofthe other. The time it takes the higher ball tofall 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 22Physics 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 movableobject over different inclinations of the plane are equalwhenever 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 oframp angle, as long as height is the same.Physics 107, Fall 2006 9Falling speedAs an object falls, it’s speed isA. ConstantB. Increasing proportional to timeC. Increasing proportional to time squaredPhysics 107, Fall 2006 10Constant acceleration• In fact, the speed of a falling object increasesuniformly with time.• We say that the acceleration is constant• Acceleration: Change in speedchange in time a =stUnits are then (meters per second)/second=(m/s)/s abbreviated m/s2Physics 107, Fall 2006 11Falling object instantaneous speedvs time• Instantaneous speedproportional to time.• So instantaneous speedincreases at a constantrate• This means constantacceleration• s=at012345670 0.1 0.2 0.3 0.4 0.5 0.6 0.7VELOCITY ( m/s)TIME ( 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 afalling object• This completelydescribes the motion• Distance proportional totime squared.00.511.522.530 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8gDISTANCE ( meters )TIME ( seconds )d = 4.9 m / s2()t2g3Physics 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 onefinger in breadth; having made this groove very straight, smooth, andpolished, and having lined it with parchment, also as smooth andpolished as possible, we rolled along it a hard, smooth, and very roundbronze ball.For the measurement of time, we employed a large vessel of water placedin an elevated position; to the bottom of this vessel was soldered a pipeof small diameter giving a thin jet of water, which we collected in asmall glass during the time of each descent... the water thus collectedwas weighed, after each descent, on a very accurate balance; thedifference and ratios of these weights gave us the differences and ratiosof the times...Using this method, Galileo very precisely determined alaw that explained the motionPhysics 107, Fall 2006 14Quantifying motion:Distance and Time• A moving objectchanges its positionwith 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 writed = s tSo knowing average speedlets us find distance traveledBUT maybe I walked0 meters in the firstsecond and then 5meters in 4 seconds.Sometimes needinstantaneous speed.Physics 107, Fall 2006 16Think about this one:You increase your speed uniformly from 0 to 60mph. This takes 6.0 seconds.Your average speed is.A. 10 mphB. 30 mphC. 40 mphD. 60 mphPhysics 107, Fall 2006 17Acceleration Acceleration is the rate at which velocity changes:Acceleration = change in velocitytime to make the changePhysics 107, Fall 2006 18Understanding accelerationZero acceleration Constant velocityconstant acceleration in the same direction as v Increasing velocityconstant acceleration opposite of v Decreasing velocity4Physics 107, Fall 2006 19Major points• position: coordinates of a body• velocity: rate of change of position–average :– instantaneous: average velocity over a very smalltime interval• acceleration: rate of change of velocity–average:– instantaneous: average acceleration over a verysmall time interval change in positionchange in time change in velocitychange in timePhysics 107, Fall 2006 20Just to check…A car’s position
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