Chapter 2Motion in One DimensionDynamics The branch of physics involving the motion of an object and the relationship between that motion and other physics concepts Kinematics is a part of dynamics In kinematics, you are interested in the description of motion Not concerned with the cause of the motionQuantities in Motion Any motion involves three concepts Displacement Velocity Acceleration These concepts can be used to study objects in motionBrief History of Motion Sumaria and Egypt Mainly motion of heavenly bodies Greeks Also to understand the motion of heavenly bodies Systematic and detailed studies Geocentric model“Modern” Ideas of Motion Copernicus Developed the heliocentric system Galileo Made astronomical observations with a telescope Experimental evidence for description of motion Quantitative study of motionPosition Defined in terms of a frame of reference A choice of coordinate axes Defines a starting point for measuring the motion Or any other quantity One dimensional, so generally the x-or y-axisDisplacement Defined as the change in position f stands for final and i stands for initial Units are meters (m) in SIf ix x x∆ ≡ −Displacement Examples From A to B xi= 30 m xf= 52 m ∆x = 22 m The displacement is positive, indicating the motion was in the positive x direction From C to F xi= 38 m xf= -53 m ∆x = -91 m The displacement is negative, indicating the motion was in the negative x directionDisplacement, GraphicalVector and Scalar Quantities Vector quantities need both magnitude (size) and direction to completely describe them Generally denoted by boldfaced type and an arrow over the letter + or – sign is sufficient for this chapter Scalar quantities are completely described by magnitude onlyDisplacement Isn’t Distance The displacement of an object is not the same as the distance it travels Example: Throw a ball straight up and then catch it at the same point you released it The distance is twice the height The displacement is zeroSpeed The average speed of an object is defined as the total distance traveled divided by the total time elapsed Speed is a scalar quantity==total distanceAverage speedtotal timedvtSpeed, cont Average speed totally ignores any variations in the object’s actual motion during the trip The total distance and the total time are all that is important Both will be positive, so speed will be positive SI units are m/sVelocity It takes time for an object to undergo a displacement The average velocity is rate at which the displacement occurs Velocity can be positive or negative ∆t is always positive−∆= =∆ −f iaveragef ix xxvt t tVelocity continued Direction will be the same as the direction of the displacement, + or - is sufficient Units of velocity are m/s (SI) Other units may be given in a problem, but generally will need to be converted to these In other systems: US Customary: ft/s cgs: cm/sSpeed vs. Velocity Cars on both paths have the same average velocity since they had the same displacement in the same time interval The car on the blue path will have a greater average speed since the distance it traveled is largerGraphical Interpretation of Velocity Velocity can be determined from a position-time graph Average velocity equals the slope of the line joining the initial and final positions An object moving with a constant velocity will have a graph that is a straight lineAverage Velocity, Constant The straight line indicates constant velocity The slope of the line is the value of the average velocityNotes on Slopes The general equation for the slope of any line is The meaning of a specific slope will depend on the physical data being graphed Slope carries unitschange in vertical axisslopechange in horizontal axis=Average Velocity, Non Constant The motion is non-constant velocity The average velocity is the slope of the straight line joining the initial and final pointsInstantaneous 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∆ →∆≡∆lim0txvtInstantaneous Velocity on a Graph The slope of the line tangent to the position-vs.-time graph is defined to be the instantaneous velocity at that time The instantaneous speed is defined as the magnitude of the instantaneous velocityAcceleration Changing velocity means an acceleration is present Acceleration is the rate of change of the velocity Units are m/s² (SI), cm/s² (cgs), and ft/s² (US Cust) −∆= =∆ −f if iv vvat t tAverage Acceleration Vector quantity When the sign of the velocity and the acceleration are the same (either positive or negative), then the speed is increasing When the sign of the velocity and the acceleration are in the opposite directions, the speed is decreasingNegative Acceleration A negative acceleration does not necessarily mean the object is slowing down If the acceleration and velocity are both negative, the object is speeding upInstantaneous and Uniform Acceleration The limit of the average acceleration as the time interval goes to zero When the instantaneous accelerations are always the same, the acceleration will be uniform The instantaneous accelerations will all be equal to the average accelerationGraphical Interpretation of Acceleration Average acceleration is the slope of the line connecting the initial and final velocities on a velocity-time graph Instantaneous acceleration is the slope of the tangent to the curve of the velocity-time graphAverage AccelerationRelationship Between Acceleration and Velocity Uniform velocity (shown by red arrows maintaining the same size) Acceleration equals zeroRelationship Between Velocity and Acceleration Velocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the same length) Velocity is increasing (red arrows are getting longer) Positive velocity and positive accelerationRelationship Between Velocity and Acceleration Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same length) Velocity is decreasing (red arrows are getting
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