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    Motion in Straight Line
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    Figure 3.21 shows the x-t plot of one-dimensional motion of a particle. Is it correct to say from the graph that the particle moves in a straight line for t < 0 and on a parabolic path for t >0 ? If not, suggest a suitable physical context for this graph.

    Position-time  graph represents the position of particle at any instant, not the trajectory of particle.

    For t < 0, the position-time graph is time axis which signifies that the particle is at rest.

    For t > 0, the position-time graph is a parabola which signifies that the body starts moving with constant acceleration.

    The given graph is a suitable context for a body is dropped from a certain height at t=0.

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    A police van moving on a highway with a speed of 30 km h-1 fires a bullet at a thief’s car speeding away in the same direction with a speed of 192 km h-1. If the muzzle speed of the bullet is 150 m s-1, with what speed does the bullet hit the thief’s car ? (Note: Obtain that speed which is relevant for damaging the thief’s car).

    Given, 


    Speed space of space the space police space van comma space straight v subscript straight p space end subscript equals space 30 space km divided by straight h space equals space 8.33 space straight m divided by straight s Muzzle space speed space of space the space bullet comma space vb space equals space 150 space straight m divided by straight s space Speed space of space the space thief ’ straight s space car comma space straight v subscript straight t space end subscript equals space 192 space km divided by straight h space equals space 53.33 space straight m divided by straight s The space he space bullet space is space fired space from space straight a space moving space van comma therefore space resultant space speed space can space be space obtained space as colon 150 space plus space 8.33 space equals space 158.33 space straight m divided by straight s Both space the space vehicles space are space moving space in space the space same space direction. space So comma space velocity space with space which space the space bullet space hits space the space thief ’ straight s space car space is comma space straight v subscript bt space equals space straight v subscript straight b space – space straight v subscript straight t space space space space space equals space 158.33 space – space 53.33 space space space space space equals space 105 space straight m divided by straight s

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    A man walks on a straight road from his home to a market 2.5 km away with a speed of 5 km h-1. Finding the market closed, he instantly turns and walks back home with a speed of 7.5 km h-1. What is the
    (a) magnitude of average velocity, and

    (b) average speed of the man over the interval of time
    (i) 0 to 30 min, (ii) 0 to 50 min, (iii) 0 to 40 min ?
    [Note: You will appreciate from this exercise why it is better to define average speed as total path length divided by time, and not as magnitude of average velocity. You would not like to tell the tired man on his return home that his average speed was zero !]
    Time space taken space by space the space man space to space reach market space from space home comma space straight t subscript 1 space equals space fraction numerator 2.5 over denominator 5 end fraction equals 1 half straight h space equals space 30 space min Time space taken space by space the space man space to space reach home space from space the space market comma space straight t subscript 2 space equals space fraction numerator 2.5 over denominator 7.5 end fraction equals 1 third straight h space equals space 20 space min Total space time space taken space in space the space whole space journey space equals space 30 space plus 20 space equals space 50 space min straight i right parenthesis thin space bold 0 bold space bold to bold space bold 30 bold space bold min Average space velocity space equals space Displaceement over time space equals space fraction numerator 2.5 over denominator begin display style bevelled 1 half end style end fraction space equals space 5 space km divided by hr Average space speed space equals fraction numerator space Distance over denominator Time space end fraction equals fraction numerator space 2.5 over denominator left parenthesis 1 divided by 2 right parenthesis end fraction space equals space 5 space km divided by straight h ii right parenthesis thin space bold 0 bold space bold to bold space bold 50 bold space bold min Time space equals space 50 space min space equals space 50 over 60 space equals space 5 divided by 6 space straight h space Net space displacement space equals space 0 Total space distance space equals space 2.5 space plus space 2.5 space equals space 5 space km Average space velocity space equals fraction numerator space Displacement over denominator Time space end fraction space equals space 0 space Average space speed space equals space fraction numerator Distance over denominator space Time end fraction space equals fraction numerator space 5 over denominator left parenthesis 5 divided by 6 right parenthesis end fraction space equals space 6 space km divided by straight h space iii right parenthesis thin space bold 0 bold space bold to bold space bold 40 bold space bold min Speed space of space the space man space equals space 7.5 space km divided by straight h Distance space travelled space in space first space 30 space min space equals space 2.5 space km Distance space travelled space by space the space man space left parenthesis from space market space to space home right parenthesis space in space the space next space 10 space min space equals space 7.5 space cross times 10 over 60 equals 1.25 space km Net space displacement space equals space 2.5 space – space 1.25 space equals space 1.25 space km Total space distance space travelled space equals space 2.5 space plus space 1.25 space equals space 3.75 space km space Average space velocity space equals space Displacment over time space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 1.25 over denominator left parenthesis 40 divided by 60 right parenthesis end fraction space equals space 1.875 space km divided by hr space Average space Speed space equals space Distance over time space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator left parenthesis 3.75 right parenthesis over denominator left parenthesis 40 divided by 60 right parenthesis end fraction space equals space 5.625 space km divided by hr
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    In Exercises 3.13 and 3.14, we have carefully distinguished between average speed and magnitude of average velocity. No such distinction is necessary when we consider instantaneous speed and magnitude of velocity. The instantaneous speed is always equal to the magnitude of instantaneous velocity. Why?

    Instantaneous velocity is given by the first derivative of distance with respect to time.

    i.e. ,     vIn = dx / dt 

    Here, the time interval dt is so small that it is assumed that the particle does not change its direction of motion.

    As a result, both the total path length and magnitude of displacement become equal is this interval of time. 

    Therefore, instantaneous speed is always equal to instantaneous velocity.
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    Explain clearly, with examples, the distinction between:
    b) The magnitude of average velocity over an interval of time, and the average speed over the same interval. [Average speed of a particle over an interval of time is
    defined as the total path length divided by the time interval].
    Show in both (a) and (b) that the second quantity is either greater than or equal to the first. When is the equality sign true ? [For simplicity, consider one-dimensional
    motion only.]

    Magnitude space of space average space velocity space equals space fraction numerator Magnitude space of space displacement over denominator Time space interval end fraction For space the space given space particle comma space Average space velocity space equals space AC over straight t Average space speed space is space given space by comma space fraction numerator Total space path space length over denominator time space interval end fraction space equals space fraction numerator AB space plus space BC over denominator straight t end fraction

    Since (AB + BC) > AC, average speed is greater than the magnitude of average velocity.

    The two quantities will be equal if the particle continues to move along a straight line.

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