Motion in One Dimension

1. Motion in OneDimension In drag racing, a driver wants as large an acceleration as possible. In a distance of one-quarter mile, a vehicle reaches speeds of more than 320 mi/h, covering the entire distance in under 5 s. (George Lepp/ Stone/Getty Images)

2. As a first step in studying classical mechanics, we describe the motion of an objectwhile ignoring the interactions with external agents that might be affecting or modifyingthat motion. This portion of classical mechanics is called kinematics. (The word kinematicshas the same root as cinema.) In this chapter, we consider only motion in one dimension,that is, motion of an object along a straight line. From everyday experience, we recognize that motion of an object represents a continuouschange in the object’s position. In physics, we can categorize motion into three types:translational, rotational, and vibrational. In our study of translational motion, we use what is called the particle model and describethe moving object as a particle regardless of its size.

3. Position, Velocity, and Speed Position: A particle’s position x is the location of the particle with respect to a chosen referencepoint that we can consider to be the origin of a coordinate system. Themotion of a particle is completely known if the particle’s position in space is known at all times. We start our clock, and once every 10 s we note the car’s position. As you can seefrom Table 2.1, the car moves to the right (which we have defined as the positivedirection) during the first 10 s of motion, from position A to position B. After B,the position values begin to decrease, suggesting the car is backing up from positionB through position F. In fact, at D, 30 s after we start measuring, the car is at theorigin of coordinates (see Fig. 2.1). It continues moving to the left and is more than50 m to the left of x = 0 when we stop recording information after our sixth datapoint.. Given the data in Table 2.1, we can easily determine the change in position ofthe car for various time intervals. The displacement Dx of a particle is defined asits change in position in some time interval. As the particle moves from an initialposition xito a final position xF its displacement is given by

4. It continues moving to the left and is more than50 m to the left of x = 0 when we stop recording information after our sixth datapoint. A graphical representation of this information is presented in Figure 2.1b.Such a plot is called a position–time graph. The average velocity vx,avgof a particle is definedas the particle’s displacement Dxdivided by the time interval Dtduring which that displacement occurs: The average velocity of a particle moving in one dimension can be positive ornegative, depending on the sign of the displacement. (The time interval Dtis alwayspositive.) If the coordinate of the particle increases in time (that is, if xf >xi), Dxis positiveand vx,avg=Dx/Dtis positive. This case corresponds to a particle movingin the positive x direction, that is, toward larger values of x. If the coordinatedecreases in time (that is, if xf<xi ), Dxis negative and hence vx,avgis negative. Thiscase corresponds to a particle moving in the negative x direction.

5. Average speed In everyday usage, the terms speed and velocity are interchangeable. In physics,however, there is a clear distinction between these two quantities. Consider a marathonrunner who runs a distance d of more than 40 km and yet ends up at herstarting point. Her total displacement is zero, so her average velocity is zero! Nonetheless,we need to be able to quantify how fast she was running. A slightly differentratio accomplishes that for us. The average speed vavgof a particle, a scalarquantity, is defined as the total distance d traveled divided by the total time intervalrequired to travel that distance:

6. Example 2.1 Calculating the Average Velocity and Speed Find the displacement, average velocity, and average speed of the car in Figure 2.1 between positions A and F.

7. This result means that the car ends up 83 m in the negative direction (to the left, in this case) from where it started. We cannot unambiguously find the average speed of the car from the data in Table 2.1 because we do not have informationabout the positions of the car between the data points. If we adopt the assumption that the details of the car’sposition are described by the curve in Figure 2.1b, the distance traveled is 22 m (from A to B) plus 105 m (from B toF), for a total of 127 m.

8. Instantaneous Velocity and Speed Often we need to know the velocity of a particle at a particular instant in time trather than the average velocity over a finite time interval Dt. In other words, youwould like to be able to specify your velocity just as precisely as you can specify yourposition by noting what is happening at a specific clock reading, that is, at some specific instant. What we have done is determine theinstantaneous velocity at that moment. In other words, the instantaneous velocity vxequals the limiting value of the ratio Dx/Dtas Dtapproaches zero:1 In calculus notation, this limit is called the derivative of x with respect to t, writtendx/dt: Instantaneous velocity 

9. The instantaneous velocity can be positive, negative, or zero. The instantaneous speed of a particle is defined as the magnitude of its instantaneousvelocity. As with average speed, instantaneous speed has no direction associatedwith it. For example, if one particle has an instantaneous velocity of 125 m/salong a given line and another particle has an instantaneous velocity of 225 m/s along the same line, both have a speed2of 25 m/s. Example X(m) t = 1 s t = 6 s x = 5 m 2 4 6 t(s) -5

10. Example A particle moves along the x axis. Its position is givenby the equation x = 3t2, with x in meters andt in seconds. Determine t=3s x =? t = 3s + Dt t = 3s the instantaneous velocity ?

11. Acceleration When the velocity of a particle changeswith time, the particle is said to be accelerating. For example, the magnitude of acar’s velocity increases when you step on the gas and decreases when you apply the brakes. Suppose an object that can be modeled as a particle moving along the x axis hasan initial velocity vxiat time tiat position A and a final velocity vxfat time tfat positionB as. The average acceleration ax,avgof the particle is defined as the changein velocity Dvxdivided by the time interval Dtduring which that change occurs: As with velocity, when the motion being analyzed is one dimensional, we can usepositive and negative signs to indicate the direction of the acceleration. Becausethe dimensions of velocity are L/T and the dimension of time is T, accelerationhas dimensions of length divided by time squared, or L/T2. The SI unit of accelerationis meters per second squared (m/s2).

12. In some situations, the value of the average acceleration may be different overdifferent time intervals. It is therefore useful to define the instantaneous accelerationas the limit of the average acceleration as Dtapproaches zero. If weimagine that point A is brought closer and closer to point B in Figure 2.6a and wetake the limit of Dvx/Dtas Dtapproaches zero, we obtain the instantaneous acceleration at point B: Instantaneous acceleration :

13. Figure 2.7 illustrates how an acceleration–time graph is related to a velocity–time graph. The acceleration at any time is the slope of the velocity–time graph atthat time. Positive values of acceleration correspond to those points in Figure 2.7awhere the velocity is increasing in the positive x direction. The acceleration reachesa maximum at time tA, when the slope of the velocity–time graph is a maximum.The acceleration then goes to zero at time tB, when the velocity is a maximum (thatis, when the slope of the vx–tgraph is zero). The acceleration is negative when thevelocity is decreasing in the positive x direction, and it reaches its most negative value at time tC.

14. From now on, we shall use the term acceleration to mean instantaneous acceleration.When we mean average acceleration, we shall always use the adjective average.Because vx=dx/dt, the acceleration can also be written as That is, in one-dimensional motion, the acceleration equals the secondderivative ofx with respect to time.

15. EXAMPLE:The velocity of a particle moving along the x axis varies according to the expression vx=40 -5t2, where vxis in meters per second and tis in seconds. Find the average acceleration in the time interval t =0 to t =2.0 s S o l u t i o n

16. PROBLEM : A particle starts from rest and accelerates as shown in Figure. Determine (a) the particle’s speed at t =10.0 s and att =15.0 s

17. Analysis Model: Particle Under Constant Acceleration If the acceleration of a particle varies in time, its motion can be complex and difficultto analyze. A very common and simple type of one-dimensional motion, however, isthat in which the acceleration is constant. In such a case, the average accelerationax,avgover any time interval is numerically equal to the instantaneous acceleration axat any instant within the interval, and the velocity changes at the same rate throughoutthe motion. This situation occurs often enough that we identify it as an analysismodel: the particle under constant acceleration. In the discussion that follows, wegenerate several equations that describe the motion of a particle for this model.

18. we can express the average velocity in any time interval as the arithmeticmean of the initial velocity vxiand the final velocity vxf: Position as a function ofvelocity and time for the particle under constant acceleration model Position as a function of time for the particle under constant acceleration model Velocity as a function of position for the particle under constant acceleration model

19. Example A jet lands on an aircraft carrier at a speed of 63 m/s. (A) What is its acceleration (assumed constant) if it stops in 2.0 s ? (B) If the jet touches down at position xi =0, what is its final position?

20. Summary

21. Analysis Models for Problem-Solving

22. An important aid to problem solving is the use of analysis models. • Analysis models are situations that we have seen in previous problems. • Each analysis model has one or more equations associated with it. Whensolving a new problem, identify the analysis model that corresponds to theproblem. The model will tell you which equations to use.