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# Chapter 2 : 1-D Kinematics: Velocity & Acceleration Hints & Answers

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Chapter 2 : 1-D Kinematics: Velocity & Acceleration Hints & Answers. –Updated 9/6/111. a) Position is the distance from a predetermined origin; displacement is the distance from the initial to the final position. The origin is arbitrary and chosen for convenience, so position vectors can vary, but displacement vectors don’t depend on the particular origin chosen in a problem.b) Instantaneous velocity is the rate of motion at one particular instant of time (as calculated from the limit of x/t =dx/dt). Average velocity is the overall displacement traveled over the total time interval thatit took. Both are the same when the velocity is constant.c) instant velocity, change in velocity and acceleration …refer to the definitions in text and/or notesd) constant velocity, constant & non-const acceleration …refer to the definitions in text and/or notese) +a is acceleration in the positive direction, -a is acceleration in the negative direction. Referring to speeds (regardless of direction), “acceleration” means the speed is increasing, while “deceleration” means the speed is decreasing. You should understand that a +a can accelerate or decelerate an object, and that a –a can also accelerate or decelerate an object. The key point is that objects “speed up” when the velocity and acceleration vectors are in the same direction (both +, or both-), and that objects “slow down” when the velocity and acceleration vectors are in opposite directions (+/- , or -/+)2. In preparation for the “Graphing Constant Acceleration” lab, sketch rough graphs of position, velocity, and acceleration vs. time for the following cases of constant acceleration: a) a=0, v>0; and also a=0, v<0; b) a>0, vo>0; and also a>0, vo<0; c) a<0, vo>0; and also a<0, vo<0 Answer these questions about the graphs you sketched: d) In which of the cases above was the speed (magnitude of velocity) increasing?...speed decreasing? e) In which of the cases above is it possible for the object turn around? f) Is it possible for the speed to be zero and the acceleration to be non-zero at any time? Explain.You will obtain these graphs during the lab “Graphing Constant Acceleration”. So come back to this problem after the lab. One of the points of the lab is to realize that speed increases (“acceleration”) when velocity and acceleration are in the same direction (+/+or -/-), whereas speed decreases (“deceleration”) when velocity and acceleration are in opposite directions (+/-or -/+).3. Fill in the missing graphs or verbal description for each case consistent with the information given. Ignore any points of abrupt change. Assume that any initial position or velocity not given is zero.a) b) c) d) e) f)VerbalDescriptionAn object is at rest some distance from theorigin, then it moves with constant velocity further away from the origin, then it stopsA car is moving with constant velocity, then it accelerates to a higher constant velocity.Object speeds up then it slows down at the same rate then it turns around and speeds up going backwards. Finallyis slows down at it returns to the origin.A mass hanging from a spring is moving up and down.An object starts from rest, it speeds up uniformly, then itmoves with constant velocity, finally it slows down uniformly to rest.A ball is tossed upwards, itrises in the air and returns to the ground.x vs. tv vs. ta vs. t a=0, except at the instants whenit starts to move or stops4. A motion diagram is an illustration showing the velocity and acceleration vectors of a moving object. Draw motion diagrams for cases (b), (c), & (e) in the problem above. a) Draw enough instantaneous velocity vectors at equal time intervals to clearly illustrate each case. b) Draw change in velocity vectors for each time interval and then draw acceleration vectors for each time interval. Ignore the points of abrupt change.Velocity ( ) and acceleration ( ) vectors (∆v vectors are directed the same as acceleration):Case (b):Case (c):Case (e):5. An equation of motion describes algebraically how the motion of an object depends on time. Write equations of motion describing cases (d), (f), & (g) above [case (e) requires 3 sets of equations]:Case (d): x(t) = A sin(t); v(t) = A cos(t) ; a(t) = -A sin(t),( where  =2π/T)Case (f): x(t) = xo + vot –gt2/2 ; v(t) = vo -gt ; a(t) = -g (up is + and down is -)Case (e) requires a separate set of formulas for each different acceleration: From 0->t1: x(t) = a1t2/2; v(t) = a1t ; a(t) = a1 From t1-> t2: x(t) = x1 + vmax t; v(t) = vmax ; a(t) = 0 From t2-> t3: x(t) = x2 + vmax t - a2t2/2; v(t) = vmax -a2t ; a(t) = -a26. When an object changes its motion abruptly in a short amount of time (as we have seen in many examples) we ignore the short time intervals for the sake of simplicity and also because these are negligible compared to the longer intervals of motion. These abrupt changes often are drawn as a sharp angle or brokenline in the motion graphs. Explain why such abrupt changes are impossible in the real world.An “abrupt change” implies a finite change in an infinitesimal amount of time, which is not possible. 7. There are two formulas that we can use to determine average velocity: vave=x/t or vave=(vi + vf)/2. a) Which one of these formulas is always correct (by definition) and which formula is only correct under certain conditions? What are the limitations of the “other” formula that is not always correct? b) A car travels 12 miles using two different constant speeds. The car travels half the distance with a speed of 10 mph and the other half distance with a speed of 30 mph. Determine the average speed here. c) The car travels the same 12 miles back but this time the driver spends half the total time of the trip traveling at 10 mph and the other half time traveling at 30 mph. Determine the average speed in the returntrip. d) Sketch position vs. time graphs for both trips. Which trip took longer (b) or (c)? e) If you changed your speed gradually from 10 mph to 30 mph during the 12 miles trip, which of the two cases above would have the same average velocity? f) Prove (or give a logical argument) that the “half-time” average velocity (c) will always be faster thanthe “half-distance” average velocity (b).a) The definition of average velocity (x/t) is always correct because it is the definition of average