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Chapter 2

Chapter 2. Motion Along a Straight Line. Motion Along a Straight Line. In this chapter we will study kinematics, i.e., how objects move along a straight line.

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Chapter 2

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  1. Chapter 2 Motion Along a Straight Line

  2. Motion Along a Straight Line • In this chapter we will study kinematics, i.e., how objects move along a straight line. • The following parameters will be defined: Displacement Average velocity Average speed Instantaneous velocity Average and instantaneous acceleration • For constant acceleration we will develop the equations that give us the velocity and position at any time. • we will study the motion under the influence of gravity close to the surface of the Earth. • Finally, we will study a graphical integration method that can be used to analyze the motion when the acceleration is not constant.

  3. Ch 2-1 Motion Along a Straight Line • Motion of an object along a straight line • Object is point mass • Motion along a horizontal or vertical or inclined (line with finite slope) line • Motion: Change in position • No change in position, body at rest

  4. Ch 2-3 Position and Displacement • Axis are used to define position of an object • Position of an object defined with respect to origin of an axis • Position x of an object is its distance from the origin at any time t • Displacement x, a vector, is change in position. • x = xf-xi • When an object changes its position, it has a velocity

  5. Here are three pairs of initial and final positions, respectively along x axis. Which pair give a negative displacement? a) -3 m, + 5 m b) -3 m, -7 m c) 7 m, -3 m Ans: x=xf-xi a) x=xf-xi=5-(-3)=+8 b) x=xf-xi=-7-(-3)=-4 c) x=xf-xi=-3-(+7)=-10 Ans: b and c Check Point 2-1

  6. Ch 2-4 Average Velocity, Average Speed • Average Velocity vavg= x/ t • vavg = (xf-xi) /(tf-ti)=Dispalcemnt/time • Average speed Savg: a scalar • Savg= total distance/ total time • Instantaneous Velocity v: • v= lim (x/ t) • t0 • Position-time graph used to define object position at any time t and to calculate its velocity • v is slope of the line on position-time graph

  7. The following equations give the position x(t) of a particle in four situations ( in each equation , x is in meters, t is in seconds, and t>0): 1) x = 3t-2 2) x=-4t2-2 3) x= 2/t2 4) x=-2 a) In which situation velocity v of the particle constant? b) In which v is in the negative direction? Ans: v=dx/dt 1) v=+3 m/s 2) v=-8t m/s 3) v = -4/t3 4) v=0 a) 1 and 4 c) 2 and 3 Check Point 2-3

  8. Ch 2-6 Acceleration • When an object changes its velocity, it undergoes an acceleration • Average acceleration aavg aavg = v/ t = (vf-vi) /(tf-ti) • Instantaneous acceleration a: a= lim (v/ t) t0 = dv/dt=d2x/dt2 • Velocity-time graph used to define object velocity at any time t and calculate its acceleration • ais slope of the line on velocity-time graph

  9. Ch 2-6 Acceleration • If the sign of the velocity and acceleration of a particle are the same, the speed of particle increases. • If the sign are opposite, the speed decreases.

  10. Check Point 2-4 • A wombat moves along an x axis. What is the sign of its acceleration if it is moving: • a) in the positive direction with increasing speed, • b) in the positive direction with decreasing speed • c) in the negative direction with increasing speed, • d) in the negative direction with decreasing speed? Ans: a) plus b) minus c) minus d) plus

  11. Ch 2-7 Constant Acceleration • Motion with constant acceleration has : • Variable Slope of position-time graph • Constant Slope of velocity -time graph • Zero Slope of acceleration -time graph • For constant acceleration a =aavg= (vf-vi)/(tf-ti) vavg= (vf+vi)/2

  12. Equations for Motion with Constant Acceleration • v=v0+at • x-x0=v0t+(at2)/2 • v2=v02+2a(x-x0) • x-x0=t(v+v0)/2 • x-x0 =vt-(at2)/2

  13. The following equations give the position x(t) of a particle in four situations: 1) x=3t-4 2) x=-5t3+4t2+6 3) x=2/t2-4/t 4) x=5t2-3 To which of these situations do the equations of Table 2-1 apply? Ans: Table 2-1 deals with constant acceleration case hence calculate acceleration for each equation: 1) a = d2x/dt2=0 2) a = d2x/dt2=-30t+8 3) a = d2x/dt2 = 12/t4-8/t2 4) a = d2x/dt2 = 10 Ans: 1 and 4 ( constant acceleration case) Check Point 2-5

  14. Ch 2-9 Free Fall Acceleration • Free fall acceleration ‘g’ dde to gravity • g directed downward towards Earth’s center along negative y-axis • with a = -g = -9.8 m/s2 • equations of motion with constant acceleration are valid for free fall motion

  15. What is the sign of the ball’s displacement for the ascent, from the release point to the highest point? B) What is it for the descent , from the highest point back to to the release point c) What is the ball’s acceleration at its highest point? Check Point 2-6 • (a) plus (upward displacement on y axis); • (b) minus (downward displacement on y axis); • (c) a = −g = −9.8m/s2

  16. Ch 2-10 Graphical Integration in Motion Analysis • Integration of v-t graph to obtain displacement xx =x-x0=vt =v dt but v dt= area between velocity curve and time axis from t0 to t • Integration of a-t graph to obtain velocity v Similarly v =v-v0= a dt • a dt = area between acceleration curve and time axis from t0 to t

  17. Old exam examples Which of the following statements is WRONG? A) A body can have constant velocity and still have a varying speed. Not Possible B) A body can have a constant speed and still have a varying velocity. uniform circular motion C) A body can have zero velocity and still be accelerating. free fall at max. height D) A body can have a constant acceleration and a variable velocity. by definition E) A body can have an upward velocity while experiencing a downward acceleration. free fall

  18. Old exam examples • FIGURE 1 shows the motion of a particle moving along an x axis with constant acceleration. What is the magnitude of the acceleration of the particle? • A) 4 m/s2 • B) 6 m/s2 • C) 3 m/s2 • D) 2 m/s2 • E) 5 m/s2 • Ans: x0=x(0)= −2 m • x−x0=v0t+ 12 at2 • x+2=v0t+ 12 at2

  19. Two cars are initially at rest. Car A is at x = 0, and car B is at x = + 600 m. They start to move, at the same time, along the same line in the positive x direction with constant accelerations: aA= 4.00 m/s2 and aB= 1.00 m/s2. How long does it take car A to overtake car B? A) 20.0 s B) 30.0 s C) 25.0 s D) 34.5 s E) 17.5 s Ans: Let d1=distance moved by A Let d2=distance moved by B When they meet: d1= d2+ 600 ⟶(1) d2= v0t+ 12aBt2=12×1.00×t2= 12t2 ⟶(3) From (2)and (3),back to (1): 2t2=12 t2+600 ⇒ 32 t2=600 t2= 23×600=400 ⇒t=20.0 s

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