AST 301Introduction to AstronomyJohn LacyRLM [email protected] LiRLM [email protected] JeonRLM [email protected] site: www.as.utexas.eduGo to Department of Astronomy courses,AST 301 (Lacy), course websiteTopics for this weekHow did Kepler improve on Copernicus’ model?State (and define the term in) Kepler’s laws.Speed, velocity, acceleration, mass, force, weightNewton’s three laws of motionNewton’s law of gravityOrbital motion explained with Newton’s lawsFalling objects explained with Newton’s lawsThe test on Friday will cover chapters 1-4, includingNewton’s lawsKepler’s laws (or rules)1. The planets move on elliptical paths with the Sun at onefocus of the ellipse for each planet.2. The speed of a planet changes during its orbit, movingfastest when it is closest to the Sun and slowest when itis farthest from the Sun. A line from the Sun to the planet sweeps out equal areasin equal times.3. Different planets move at different speeds, with a planetin a smaller orbit moving faster than one in a larger orbit. The time for a planet to orbit the Sun depends on thesize of it orbit according to the rule:P2 = a3, where P is in years and a is in AUKepler’s 2nd law and proportionsOne way of stating Kepler’s 2nd law is to say:The speed of a planet at different places in its orbit isinversely proportional to its distance from the Sun.This means that if it is twice as far from the Sun in onelocation than another, it moves ½ as fast at that location.v α 1/rWe will be able to state Kepler’s 3rd law in a similar way:The average speed of a planet (when we compare it to adifferent planet) is inversely proportional to the squareroot of its average distance from the Sun (a).v α 1/r1/2Galileo’s contributions to scienceIn addition to his support of the Copernican model of theUniverse, Galileo argued that science should be basedon observations and experiments, not on pure thought.In his book, “Dialogues on Two New Sciences” he beganthe modern study of motion and materials.Galileo’s most famous experiment was dropping two ballsoff of the leaning tower of Pisa.Most people thought the heavier ball would fall faster.What actually happened?About 100 years later, Newton continued Galileo’s study ofmotion (and other sciences).Newton’s theory of motionCommon experience tells us that you have to push on anobject to make it move and the harder you push thefaster it goes.If you stop pushing on an object it will stop moving.Newton realized that this is a result of a complicatedsituation involving friction.In a simple situation (like an object in space or a verysmooth ball rolling) the rules are different.And the rules for simple situations can be combined toexplain motion in complicated situations.Newton also realized that his laws of motion when appliedto the planets could explain Kepler’s laws.Definitionsspeed: distance traveled / time spent travelingacceleration: change of speed / time spent changing(can also be a change in direction of motion)force: how hard one object pushes or pulls on anothermass: is a measure of the amount of matter in an objectand how much it resists being pushed aroundvolume: the amount of space an object takes upfor a box: height x width x depthNewton’s laws of Motion1. If there are no forces acting on an object, if moving it willcontinue to move with constant speed and in the samedirection. If not moving, it will remain stationary.2. If a force is acting on an object, the object’s acceleration(the rate of change in its speed or direction of motion) isproportional to the force acting on it and is inverselyproportional to its mass.a α F / m3. If an object exerts a force on a second object, thesecond object necessarily exerts the same force on thefirst, but in the opposite direction.QuizIf I step on my brakes and slow down from 60 mph to 30mph, am I accelerating?A. yesB. noBy scientists’ definition any change in speed is calledacceleration, whether an increase or a decrease.If I take my foot off of the gas and coast around a corner atconstant speed, am I accelerating?A. yesB. noQuizIf I push on my car for 5 seconds I can make it move at 2mph. If I pushed for an additional 5 seconds with thesame force, how fast would it then be going? Ignore allother forces acting on the car.A. 2 mphB. 4 mphC. 7 mphD. 10 mphQuizIf I push on my car for 5 seconds I can make it move at 2mph. If two identical cars were lined up bumper-to-bumper, how fast could I make them move by pushing(with the same force) for 5 seconds?A. 0 mphB. 1 mphC. 2 mphD. 4 mphAgain, ignore all other forces acting on the cars.Newton’s law of gravityGravity is a attraction between any two objects.The force of gravity is proportional to the product of themasses of the two objects and is inversely proportionalto the distance between their centers.F α m1 m2 / d2orF = G m1 m2 / d2QuizWhen the Apollo astronauts were on the way to the Moon,10 Earth radii away from the Earth (1/6 of the way to theMoon), what was the force of gravity on a 150 poundastronaut?A. 0B. 1.5 lbC. 15 lbD. 150 lbNewton’s explanation of Galileo’s experimentIf the two balls that Galileo dropped from the leaning towerhad different masses, the force of gravity on them shouldhave been different. Why did they fall at the same rate?What if one ball had a mass of 1 kilogram and the otherhad a mass of 2 kg? The force of gravity on the 2 kg ballwas twice the force on the 1 kg ball and Newton’s 2nd lawsays that acceleration is proportional to the force on anobject.But Newton’s 2nd law also says that acceleration isinversely proportional to the mass of the object.Doubling the force and doubling the mass have oppositeeffects, resulting in the same acceleration for the twoballs.Galileo (and Einstein) just argued that all objects acceleratethe same under the influence of gravity. Newtonexplained that by saying that the force of gravity isproportional to mass, and acceleration is proportional toforce / mass.If the force of the Earth’s gravity on a 1 kg ball is 10Newtons, what is the force of the ball’s gravity on theEarth?A. much less than 1 Nt.B. 1 NtC. 10 Nt.D. much more than 10 Nt.Newton’s version of Kepler’s 3rd lawNewton’s laws can be used to derive Kepler’s 3rd law.A force is needed to cause a planet to move on a curvedpath. That force is the force of gravity of the Sun.By equating the
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