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Unformatted text preview: Chapter 2 Motion in One Dimension 2.1 The Important Stuff 2.1.1 Position, Time and Displacement We begin our study of motion by considering objects which are very small in comparison to the size of their movement through space. When we can deal with an object in this way we refer to it as a particle . In this chapter we deal with the case where a particle moves along a straight line. The particle’s location is specified by its coordinate , which will be denoted by x or y . As the particle moves, its coordinate changes with the time, t . The change in position from x 1 to x 2 of the particle is the displacement Δ x , with Δ x = x 2 x 1 . 2.1.2 Average Velocity and Average Speed When a particle has a displacement Δ x in a change of time Δ t , its average velocity for that time interval is v = Δ x Δ t = x 2 x 1 t 2 t 1 (2.1) The average speed of the particle is absolute value of the average velocity and is given by s = Distance travelled Δ t (2.2) In general, the value of the average velocity for a moving particle depends on the initial and final times for which we have found the displacements. 2.1.3 Instantaneous Velocity and Speed We can answer the question “how fast is a particle moving at a particular time t ?” by finding the instantaneous velocity . This is the limiting case of the average velocity when the time 27 28 CHAPTER 2. MOTION IN ONE DIMENSION interval Δ t include the time t and is as small as we can imagine: v = lim Δ t → Δ x Δ t = dx dt (2.3) The instantaneous speed is the absolute value (magnitude) of the instantaneous ve locity. If we make a plot of x vs. t for a moving particle the instantaneous velocity is the slope of the tangent to the curve at any point. 2.1.4 Acceleration When a particle’s velocity changes, then we way that the particle undergoes an acceleration . If a particle’s velocity changes from v 1 to v 2 during the time interval t 1 to t 2 then we define the average acceleration as v = Δ x Δ t = x 2 x 1 t 2 t 1 (2.4) As with velocity it is usually more important to think about the instantaneous accel eration , given by a = lim Δ t → Δ v Δ t = dv dt (2.5) If the acceleration a is positive it means that the velocity is instantaneously increasing ; if a is negative, then v is instantaneously decreasing . Oftentimes we will encounter the word deceleration in a problem. This word is used when the sense of the acceleration is opposite that of the instantaneous velocity (the motion). Then the magnitude of acceleration is given, with its direction being understood. 2.1.5 Constant Acceleration A very useful special case of accelerated motion is the one where the acceleration a is constant. For this case, one can show that the following are true: v = v + at (2.6) x = x + v t + 1 2 at 2 (2.7) v 2 = v 2 + 2 a ( x x ) (2.8) x = x + 1 2 ( v + v ) t (2.9) In these equations, we mean that the particle has position x and velocity v at time t = 0; it has position x and velocity v at time t ....
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 Spring '10
 Carter
 mechanics, Acceleration, Velocity, ball

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