1Mon. Jan 23 Physics 107, Spr06 1From last time…Inertia:tendency of body to continue in straight-line motion atconstant 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 of falling distance.• But WHY?• Gravity disturbs the object, leading to falling motion.But how does this lead to Galileo’s results?Mon. Jan 23 Physics 107, Spr06 2Position, speed, velocity, &acceleration• Need to understand these concepts– position– distance– speed, velocity• average• instantaneous– acceleration• average• instantaneousMon. Jan 23 Physics 107, Spr06 3Quantifying 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 motionMon. Jan 23 Physics 107, Spr06 4• If I move 5 meters in 5 seconds.Then # meters in each second =5 divided by 5 = 1 meter per second.Can use this information to find the speedBUT maybe I walked 0 meters in the first second and then 5 meters in 4 seconds.e.g. could walk 1 meter in the first second, and1 meter in the next second, etc.Mon. Jan 23 Physics 107, Spr06 5The average speed is the same ! Average speed = distance traveledtraveling timeAs an equation: ! Distance traveled = dTraveling time = tAverage speed = s ! s =dtCould also write! d = s tSo knowing average speedlets us find distance traveledMon. Jan 23 Physics 107, Spr06 6Instantaneous speedInstantaneous speed is the average velocityover an infinitesimal (very short) timeinterval.This is what your speedometer reads.Instantaneous speed gives you a betterunderstanding of the motion.2Mon. Jan 23 Physics 107, Spr06 7Think 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 mphMon. Jan 23 Physics 107, Spr06 8Acceleration ! Acceleration is the rate at which velocity changes :Acceleration = change in velocitytime to make the changeMon. Jan 23 Physics 107, Spr06 9Understanding accelerationZero acceleration ⇒ Constant velocityconstant acceleration in the same direction as v ⇒ Increasing velocityconstant acceleration opposite of v ⇒ Decreasing velocityMon. Jan 23 Physics 107, Spr06 10Major 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 timeMon. Jan 23 Physics 107, Spr06 11Just to check…A car’s position on a highwayis plotted versus time.It turns out to be a straight line.Which of these statements is true?A. Its acceleration is negativeB. Its acceleration is positiveC. Its acceleration is zeroD. Its velocity is zeroTime (s)Position (m)Mon. Jan 23 Physics 107, Spr06 12Why a=0?• Position vs time is a straight line… x = vt• …means constant velocity (= )• Constant velocity means zero accelerationTime (s)Position (m)Change in positionChange in time ! change in positionchange in timeConstant velocityZero acceleration3Mon. Jan 23 Physics 107, Spr06 13Constant accelerationWhat about constant acceleration?• Acceleration =• Constant acceleration:– For every time interval (say, 1 second), thevelocity chnages by the same amount.– a>0 gives a uniformly increasing velocity: v = at ! change in velocitychange in timeTime (s)Velocity (m/s)Constant accelerationMon. Jan 23 Physics 107, Spr06 14QuestionYou are traveling at 60 miles per hour.You apply the brakes, resulting in a constantnegative acceleration of -10 mph / second.How many seconds does it take to stop?A. 10 secondsB. 6 secondsC. 3 secondsVelocity change is 10 mph forevery second. Takes six seconds todecrease the velocity to zeroMon. Jan 23 Physics 107, Spr06 15QuestionsHow far does the car go during that time?A. 0.1 mileB. 0.2 mileC. 0.05 mileSince speed changes uniformly with time(from 60 mph to 0 mph), so average speed is 30 mph.Distance = average speed x time = (30 miles/hour) x (6 seconds) = = (30 miles/hour) x (1/600 hr) = 1/20 mileMon. Jan 23 Physics 107, Spr06 16Back to Galileo• Use position, velocity,acceleration to quantifythe motion of a fallingobject.Mon. Jan 23 Physics 107, Spr06 17Distance vs time for falling ball• From analyzing thevideo frame by framewe find the position vstime.• 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 0.8Falling BallDISTANCE ( meters )TIME ( seconds )! d = 4.9 m /s2( )t2Mon. Jan 23 Physics 107, Spr06 18Average speed for falling ball• Total time=0.73s• Total distance=2.6m• Avg speed =2.6m/0.73s=3.5!m/s00.511.522.530 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Falling BallDISTANCE ( meters )TIME ( seconds )4Mon. Jan 23 Physics 107, Spr06 19Instantaneous speed00.511.522.530 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Falling BallDISTANCE ( meters )TIME ( seconds ) ! speed =0.4m " 0.2m0.275s " 0.18s= 2.1m / s0.22s ! speed =1.2m " 0.8m0.488s " 0.40s= 4.5m / s0.45s ! speed =2.6m " 2.0m0.73s " 0.638s= 6.5m / s0.69sMon. Jan 23 Physics 107, Spr06 20Instantaneous speed vs time• Instantaneous speedproportional to time.• So instantaneous speedincreasesat a constant rate• 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=6.8 m / s0.69 s= 9.85 m / s /s = 9.85 m / s2Mon. Jan 23 Physics 107, Spr06 21Uniform acceleration from restAcceleration = constant = a = 9.8 m/s2Velocity = (acceleration)x(time)= atUniformly increasing velocityDistance = (average vel)x(time)= (1/2)at x t = (1/2)at2Mon. Jan 23 Physics 107, Spr06 22Falling object: constant acceleration• The data show that a falling object has constantacceleration.• This is called the acceleration of gravity9.8 m/s/s = 9.8 m/s2• But why does gravity result in a constantacceleration?• Why is this acceleration independent of mass?Mon. Jan 23 Physics 107, Spr06 23Tough questions• These are difficult questions.Maybe not
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