1 1 Solve 1 2 Solve 1 3 Solve 1 4 Solve a The basic idea of the particle model is that we will treat an object as if all its mass is concentrated into a single point The size and shape of the object will not be considered This is a reasonable approximation of reality if i the distance traveled by the object is large in comparison to the size of the object and ii rotations and internal motions are not significant features of the object s motion The particle model is important in that it allows us to simplify a problem Complete reality which would have to include the motion of every single atom in the object is too complicated to analyze By treating an object as a particle we can focus on the most important aspects of its motion while neglecting minor and unobservable details b The particle model is valid for understanding the motion of a satellite or a car traveling a large distance c The particle model is not valid for understanding how a car engine operates how a person walks how a bird flies or how water flows through a pipe a An operational definition defines a concept or an idea in terms of a procedure or a set of 1 5 Solve operations that is used to identify or measure the concept b The displacement its final location Mathematically cid 71 r of an object is a vector found by drawing an arrow from the object s initial location to cid 71 of an object is a vector that points in the The average velocity v same direction as the displacement interval during which the object moves from its initial location to its final location and has length or magnitude cid 71 r t where is the time t t t i f cid 71 i r cid 71 r cid 71 r f cid 71 r 1 6 Solve The player starts from rest and moves faster and faster accelerates 1 7 Solve The particle starts with an initial velocity but as it slides it moves slower and slower till coming to rest This is a case of negative acceleration because it is an acceleration opposite to the positive direction of motion cid 71 1 8 Solve The acceleration of an object is a vector formed by finding the ratio of v object s velocity to of cid 71 t the time in which the change occurs The acceleration vector a which is found by vector subtraction cid 71 v the change in the points in the direction a Acceleration is found by the method of Tactics Box 1 3 Let be the velocity vector cid 71 0v 1 9 Solve cid 71 between points 0 and 1 and 1v be the velocity vector between points 1 and 2 0v because more distance is covered in the same interval of time b Speed 1v is greater than speed 1 10 Solve cid 71 between points 0 and 1 and 1v be the velocity vector between points 1 and 2 a Acceleration is found by the method of Tactics Box 1 3 Let be the velocity vector cid 71 0v b Speed 1v is greater than speed 0v because more distance is covered in the same interval of time 1 11 Solve a b 1 12 Solve a b 1 13 Model Represent the car as a particle Visualize The dots are equally spaced until brakes are applied to the car Equidistant dots indicate constant average speed On braking the dots get closer as the average speed decreases 1 14 Model Represent the child sled system as a particle Visualize The dots in the figure are equally spaced until the sled encounters a rocky patch Equidistant dots indicate constant average speed On encountering a rocky patch the average speed decreases and the sled comes to a stop This part of the motion is indicated by a separation between the dots that becomes smaller and smaller 1 15 Model Represent the tile as a particle Visualize The tile falls from the roof with an acceleration equal to a g 9 8 m s2 Starting from rest its velocity increases until the tile hits the water surface This part of the motion is represented by dots with increasing separation indicating increasing average velocity After the tile enters the water it settles to the bottom at roughly constant speed 1 16 Model Represent the tennis ball as a particle Visualize The particle falls freely for the three stories under the acceleration of gravity It strikes the ground and very quickly decelerates to zero while decompresses and finally travels upward with negative acceleration under gravity to zero velocity at a height of two stories The downward and upward motions of the ball are shown in the figure The increasing length between the dots during downward motion indicates increasing average velocity or downward acceleration On the other hand the decreasing length between the dots during upward motion indicates acceleration in a direction opposite to its motion that is in the downward direction Assess For a free fall motion acceleration due to gravity is always vertically downward 1 17 Model Represent the toy car as a particle Visualize As the toy car rolls down the ramp its average speed increases This is indicated by the increasing length of the velocity arrows That is motion down the ramp is under an acceleration At the bottom of the ramp the toy car continues with the speed obtained with no change in velocity cid 71 a b 1 18 Solve a Dot 1 2 3 4 5 6 7 8 9 Time s 0 2 4 6 8 10 12 14 16 x m 0 30 95 215 400 510 600 670 720 1 19 Solve A forgetful physics professor goes for a walk on a straight country road Walking at a constant speed he covers a distance of 300 m in 300 s He then stops and watches the sunset for 100 s Finding that it was getting dark he walks faster back to his house covering the same distance in 200 s 1 20 Solve Forty miles into a car trip north from his home in El Dorado an absent minded English professor stopped at a rest area one Saturday After staying there for one hour he headed back home thinking that he was supposed to go on this trip on Sunday Absent mindedly he missed his exit and stopped after one hour of driving at another rest area 20 miles south of El Dorado After waiting there for one hour he drove back very slowly confused and tired as he was and reached El Dorado in two hours 1 21 Visualize The bicycle is moving with an acceleration of 1 5 m s2 Thus the velocity will increase by 1 5 m s each second of motion cid 71 1 22 Visualize The particle moves upward with a constant acceleration a and is reached at a height of 1000 m The final velocity is 200 m s 1 23 Solve a 9 12 s 9 12 s 9 12 …
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