Describes motion while ignoring the agents that caused the motion For now, will consider motion in one dimension

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1 Kinematics Describes motion while ignoring the agents that caused the motion For now, will consider motion in one dimension Along a straight line Will use the particle model A particle is a point-like object, has mass but infinitesimal size

2 Position The object s position is its location with respect to a chosen reference point Consider the point to be the origin of a coordinate system In the diagram, allow the road sign to be the reference point

3 Position-Time Graph The position-time time graph shows the motion of the particle (car) The smooth curve is a guess as to what happened between the data points

4 Motion of Car Note the relationship between the position of the car and the points on the graph Compare the different representations of the motion

5 Data Table The table gives the actual data collected during the motion of the object (car) Positive is defined as being to the right

6 Alternative ti Representations ti Using alternative representations is often an excellent strategy for understanding a problem For example, the car problem used multiple representations Pictorial representation Graphical representation Tabular representation Goal is often a mathematical representation

7 Displacement Defined as the change in position during some time interval Represented as x x x f - x i SI units are meters (m) x can be positive or negative Different than distance the length of a path followed by a particle

8 Distance vs. Displacement An Example Assume a player moves from one end of the court to the other and back Distance is twice the length of the court Distance is always positive Displacement is zero Δx = x f x i = 0 since x f = x i

9 Average Velocity The average velocity is rate at which the displacement occurs x x f x i v xavg, t t The x indicates motion along the x-axis The dimensions are length / time [L/T] The SI units are m/s Is also the slope of the line in the position time graph

10 Average Speed Speed is a scalar quantity same units as velocity total distance / total time: v avg d t The speed has no direction and is always expressed as a positive number Neither average velocity nor average speed gives details about the trip described

11 Instantaneous t Velocity The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero The instantaneous velocity indicates what is happening at every point of time

12 Instantaneous t Velocity, graph The instantaneous velocity is the slope of the line tangent to the x vs. t curve This would be the green line The light blue lines show that as t gets smaller, they approach the green line

13 Instantaneous Velocity, equations The general equation for instantaneous velocity is v x lim0 t x x dx t dt The instantaneous velocity can be positive, negative, or zero

14 Instantaneous t Speed The instantaneous speed is the magnitude of the instantaneous velocity The instantaneous t speed has no direction associated with it

15 Vocabulary Note Velocity and speed will indicate instantaneous values Average will be used when the average velocity or average speed is indicated

16 Particle Under Constant Velocity Constant velocity indicates the instantaneous velocity at any instant during a time interval is the same as the average velocity during that time interval: RECALL x xf xi v xavg, t t v x = v x, avg =CONSTANT! HERE The mathematical representation of this situation is the equation x x x v f i or x x v t x f i x t t Common practice is to let t i = 0 and v x =v for one direciton and the equation becomes: x f = x i + v t (for constant v in one dimension)

17 Particle Under Constant Velocity, Graph The graph represents the motion of a particle under constant velocity The slope of the graph is the value of the constant velocity The y-intercept is x i

18 Average Acceleration Acceleration is the rate of change of the velocity a xavg, Dimensions are L/T 2 SI units are m/s² v v v t t t x xf xi In one dimension, positive and negative can be used to indicate direction f i

19 Instantaneous t Acceleration The instantaneous acceleration is the limit of the average acceleration as t approaches 0 a x lim x x t 0 t dt dt 2 v dv d x The term acceleration will mean instantaneous acceleration If average acceleration is wanted, the word average will be included 2

20 Instantaneous Acceleration -- graph The slope of the velocity-time graph is the acceleration The green line represents the instantaneous acceleration The blue line is the average acceleration

21 Graphical Comparison Given the displacementtime graph (a) The velocity-time graph is found by measuring the slope of the position-time graph at every instant The acceleration-time graph is found by measuring the slope of the velocity-time graph at every instant

22 Graphical Representations of Motion Acceleration and Velocity Observe the graphs of the car under various conditions describe the motion (ie a and v?) Note the relationships among the graphs Set various initial velocities, positions and accelerations

23 4 Kinematic Equations 1D with constant t acceleratione at derived via calculus and algebra 1. Given dv/dt =a -> dv= a dt integrate( t =0 to t v= v 0 to v) Gives v = v 0 + at 2. given v= dx/dt =v 0 + at integrate( t =0 to t x= x 0 to x) Gives x=x x +1/2at 2 0 +v 0 t 3, For problem solving it is useful to eliminate a by substituing for a from 1 Into 2 Gives: x=x x 0 +1/2(v 0 + v)t 4. For problem solving eliminate t in 2 by substituting for t from 1 Gives: v 2 = v a (x-x 0 ) You may need to use two of the equations to solve one problem

24 Test Graphical Interpretations t ti Match a given velocity graph with the corresponding acceleration graph Match a given acceleration graph with the corresponding velocity graph(s)

25 Galileo Galilei i Italian physicist and astronomer Formulated laws of motion for objects in free fall Supported heliocentric universe

26 Freely Falling Objects A freely falling object is any object moving freely under the influence of gravity alone. It does not depend upon the initial motion of the object Dropped released from rest Thrown downward Thrown upward

27 Acceleration of all Freely Falling Object found by Galileo The acceleration of an object in free fall is directed downward, regardless of the initial motion The magnitude of free fall acceleration is g = 9.80 m/s 2 NOTE: magnitude means just value not direction! g decreases with increasing altitude g varies with latitude 980m/s is the average at the Earth s surface The symbol g is normally used for the acceleration due to gravity ie a=g=-9.80 m/s 2 on average Not to be confused with g for grams

28 Free Fall an object dropped d Initial velocity is zero Let up be positive Use the kinematic equations Generally use y instead of x since vertical Acceleration is a y = -g = m/s 2 v o = 0 a = -g

29 Free Fall an object thrown downwardd a g 980m/s y = -g = Initial velocity 0 With upward being positive, initial velocity will be negative v o 0 a = -g

30 Free Fall -- object thrown upward Initial velocity is upward, so positive The instantaneous velocity at tthe maximum height htis zero v = 0 a y = -g = m/s 2 v o 0 y everywhere in the motion a = -g

31 Free Fall Example Initial velocity at A is upward (+) and acceleration is -g (-9.8 m/s 2 ) At B, the velocity is 0 and the acceleration 2 is -g (-9.8 m/s 2 ) At C, the velocity has the same magnitude as at A, but is in the opposite direction The displacement is 50.0 m (it ends up 50.0 m below its starting point)

32 But motion here is in the y direction so for this image the equations can be written As below. Still 1 dimension but y is the direction here! Initial in now i rather than 0 f for final etc. see equivalent 1 D equations in red. v yf = v yi +a y t v = v 0 + at y f =y i + ½ (v yi + v yf )t : x=x 0 +1/2(v 0 + v)t y f = y i + v yi t + ½ a y t 2 x=x 0 +v 0 t +1/2 a t 2 v yf2 = v yi2 + 2a y (y f -y i ) : v 2 =v a (x-x 0 ) Some books use these with a y =-g-> a=-g use the following for clarity and simplicity in the y direction (can you use these to v = v 0 - gt get the values in the y=y y 0 +1/2(v 0 + v)t figure to the left? Try it) y=y 0 +v 0 t -1/2 g t 2 v 2 =v g (y-y 0 )

33 Problem Solving Finalize Check your result Does it have the correct units? Does it agree with your conceptualized ideas? Does it agree with your conceptualized ideas? Look at limiting situations to be sure the results are reasonable Compare the result with those of similar problems

34 HOMEWORK Physics IIA Simple drill and practice with the 4 kinematic equations. Use these to solve the following Be sure to organize what you know and what you are looking for to help your logic. HW 16.Given initial velocity, v o, is 10 cm/sec and final velocity, v, is 100 cm/sec and The time to accelerate( constant) was 4 seconds than what was the acceleration,a? HW 17. Given an initial velocity in an auto race was 50 km/hr at a starting position of 100 meters from the starting place (x=0), The car accelerates from 50 to 150 km/hr in 6 seconds. Watch your units, convert to the same units How far down the track in meters (assume linear) is the car after the 6 seconds? HW 18. A cigarette boat (fast racer!) accelerates at 60 m/sec 2 from 0 initial velocity for 5 seconds a. How fast is it going at the end of 5 seconds, in m/sec, in km/hr, in miles/hr? b. How far has it traveled in this time? HW 19. Kinematics in the vertical direction. You shoot an arrow straight up. It leaves your bow at 160 km/hr. (watch units) a. How far up does it go before it starts down.? b. How long does it take to get to the top of its motion?

35 Homework PHYSICS IIA continued HW 20. Algebraically derive equations 3 and 4 above as described on the slide HW 21.. A truck covers 40.0 m in 8.50 s while smoothly slowing down to a final speed of 2.80 m/s. a. Find its original i speed b. Find its acceleration NOTE it must be<0 HW 22 A jet plane comes in for a landing with a speed of 100m/s. and its acceleration Can have a maximum magnitude of 5.00 m/s 2 as it comes to rest. NOTE also <0 a. From the instant the plane touches the runway, what is the minimum time interval needed before it can come to rest? b. Can this plane land on a small tropical island airport where he runway is km long? c. Explain your answer with calculations.? HW 23 A baseball is hit so that it travels straight upward. A fan observers it take 3.00 s To reach its maximum height. Find a. The ball s initial velocity?.hint. What is the velocity at the top of the motion? b. The height it reaches?

36 Sir Isaac Newton Formulated basic laws of mechanics Discovered Law of Universal Gravitation Invented form of calculus Many observations dealing with light and optics

37 Force Forces are what cause any change in the velocity of an object Newton s definition Force =dp/dt p = mv defined at the momentum of mass m A force is that which causes an acceleration Ie if m is constant than F=dp/dt=d(mv)/dt=mdv/dt And dv/dt =a so F=ma in this case or a=f/m This latter connects the kinematics we did to dynamics of forces

38 Classes of Forces Contact forces involve physical contact between two objects Examples a, b, c Field forces act tthrough hempty space Also called action at a distance! No physical contact is required Examples d, e, f MAGIC? Not touching? We set up a space containing the FIELD Force direction and strengths for a give situation (baseball field=place of baseball So magnetic field =?)

39 Fundamental Forces Gravitational force Between objects Electromagnetic forces Between electric charges Nuclear force Between subatomic particles Weak forces Arise in certain radioactive decay processes Note: These are all field forces

40 More About Forces A spring can be used to calibrate the magnitude of a force Doubling the force causes double the reading on the spring When both forces are applied, the reading is three times the initial reading

41 Newton s First Law This is also called the law of inertia In the absence of external forces, when viewed from an inertial reference frame, an object at rest remains at rest (a=0) and an object in motion continues in motion with a constant velocity NOTE: a reference frame in which the object has zero acceleration and does not interact with other objects is called an inertial reference frame A reference frame that moves with constant velocity relative to the distant stars is the best approximation of an inertial frame We can consider the Earth to be such an inertial frame, Although it has a small centripetal acceleration associated with its motion. That is, rotation..so life s not perfect

42 Inertia and Mass The tendency of an object to resist any attempt to change its velocity is called inertia Mass is that property of an object that specifies how much resistance an object exhibits to changes in its velocity Masses can be defined in terms of the accelerations produced by a given force acting on them: m a 1 2 m 2 1 The magnitude of the acceleration acting on an object is inversely yproportional p to its mass (or a=f/m 2 nd law!) a

43 More About Mass Mass is an inherent property of an object Mass is independent of the object s surroundings Mass is independent of the method used to measure it Mass is a scalar quantity The SI unit of mass is kg

44 Mass vs. Weight Mass and weight are two different quantities Weight is equal to the magnitude of the gravitational ti force exerted on the object Weight will vary with location Example: w earth = 180 lb; w moon ~ 30 lb m earth = 2 kg; m moon = 2 kg Weight and Gravitational Mass

45 Newton s Second Law When viewed from an inertial reference frame, the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass Force is the cause of change in motion, as measured by the acceleration Algebraically, F a F ma m With a proportionality constant of 1 and speeds much lower than the speed of light

46 Units of Force The SI unit of force is the Newton (N) or Dyne (CGS) 1N = 1k kg m / s 2 OR 1 Dyne =1gm cm/s 2 The US Customary unit of force is a pound (lb) 1 lb = 1 slug ft / s 2 1 N ~ ¼ lb

47 Gravitational ti Force The gravitational force, F g, is the force that the earth exerts on an object This force is directed toward the center of the earth From Newton s Second Law and the fact that all objects accelerate at g to the ground Then since the Force=ma F g m g Its magnitude is called the weight of the object Weight = F g = mg

48 More About Weight Because it is dependent on g, the weight varies with location g, and therefore the weight, is less at higher altitudes This can be extended to other planets, but the value of g varies from planet to planet, so the object s weight will vary from planet to planet Weight is not an inherent property of the object

49 Gravitational Mass vs. Inertial Mass In Newton s Laws, the mass is the inertial mass and measures the resistance to a change in the object s motion In the gravitational force, the mass is determining the gravitational attraction between the object and the Earth Experiments show that gravitational mass and inertial mass have the same value

50 Newton s Third Law If two objects interact, the force F 12 exerted by object 1 on object 2 is equal in magnitude and opposite in direction to the force F exerted by 21 object 2 on object 1 Note on notation: is the force exerted by A on B F F F AB Newton s Third Law

51 Newton s Third Law, Alternative ti Statements t t Forces always occur in pairs A single isolated force cannot exist The action force is equal in magnitude to the reaction force and opposite in direction One of the forces is the action force, the other is the reaction force It doesn t matter which is considered the action and which the reaction The action and reaction forces must act on different objects and be of the same type

52 Action-Reaction Examples, 1 F F 12 The force exerted by object 1 on object 2 is equal in magnitude and opposite in direction to F exerted 21 by object 2 on object 1 F F The Normal Force

53 Action-Reaction Examples, 2 The normal force (table on monitor) is the reaction of the force the monitor exerts on the table Normal means perpendicular, in this case The action (Earth on monitor) force is equal in magnitude and opposite in direction to the reaction force, the force the monitor exerts on the Earth

54 Free Body Diagram In a free body diagram, you want the forces acting on a particular object Model the object as a particle The normal force and the force of gravity are the forces that act on the monitor

55 Particles in Equilibrium i If the acceleration of an object that can be modeled as a particle is zero, the object is said to be in equilibrium The model is the particle in equilibrium model Mathematically, the net force acting on the object is zero we note sum of all components in all F 0 directions must be 0! this is the key to solving static problems F x 0and F 0 y

56 Equilibrium, i Example 1a A lamp is suspended from a chain of negligible mass The forces acting on the lamp are the downward d force of gravity the upward tension in the chain Applying equilibrium gives Fy 0 T Fg 0 T Fg

57 Equilibrium, i Example 2a Example 5.4 Conceptualize the traffic light Assume cables don t break Nothing is moving Categorize as an equilibrium problem No movement, so acceleration is zero Model as a particle in equilibrium

58 Newton s Second Law, Example 1a Forces acting on the crate: A tension, acting through the rope, is the magnitude of force T F The gravitational force, The normal force, n, exerted by the floor F g

59 Newton s Second Law, Example 1b Apply Newton s Second Law in component form: F T ma x Solve for the unknown(s) x F n F 0 n F y g g If the tension is constant, then a is constant and the kinematic equations can be used to more fully describe the motion of the crate

60 Note About the Normal Force The normal force is not always equal to the gravitational force of the object For example, in this case n F n F F y and may also be less than g n F F g 0 F g

61 Home work PHYSICS IIB FUN WITH NEWTONS LAWS HW 24. one Newton =? Dynes? HW 25. To accelerate a 100 kg mass to 40 m/s 2 one needs a force of? A.In Newtons? B. convert to DYNES! HW 26. The forces on you standing still on the floor are? And what is the net force? HW 27 In the hanging lamp slide the mass of the lamp is 8kg what is the tension in The chain holding it. HW 28 electron of mass 9.11 x kg had an initial speed of 3.00 x 10 2 m/s. I It travels in a straight line, and its speed increases to 7.00 x 10 5 m/s in a distance of 5.0 cm, assume a is a constant. (NOTE: watch units keep them the same in the equations used) a. Determine the force exerted on the electron? b. Compare the force to the weight of the electron? (which was ignored above) HW 29. An airplane needs to lift off a 500 m runway. It starts from rest and must reach a speed equivalent to 120 miles/hr at the 500 m point(watch units) to lift off the runway. a. What acceleration must it obtain to safely leave. (assume constant) b. Assuming the thrust or Force to obtain the necessary acceleration is from a single Jet engine..how much Force (assume constant) must this engine provide to lift the plane off the runway. EXTRA CREDIT: in the traffic light slide, the light is 50 kg what are the tensions,t 1,T 2,T 3 HINtT: look at Example 1a

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