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Calculate change in KE and momentum (impulse)

  1. Aug 7, 2017 #1
    1. The problem statement, all variables and given/known data

    I have attached the problem.

    A 14,000 kg tractor traveling north at 21 km/h turns west and travels 26 km/h. Calculate the change in the tractor's

    a. kinetic energy
    b. linear momentum (magnitude AND direction)

    2. Relevant equations

    delta KE = (1/2)mv^2 - (1/2)mvi^2

    change in linear momentum (delta p) = m(vf-vi)

    3. The attempt at a solution

    I defined east as positive and north as positive.

    I first converted the velocities to m/s to get: 5.83 m/s North and -7.22 m/s West.

    a) I used: delta KE = (1/2)mv^2 - (1/2)mvi^2

    Since kinetic energy is a scalar quantity, I do not have to worry about direction. So... I will just plug in the values all given to find the change in KE.

    (1/2)(14000)(-7.22)^2 - (1/2)(14000)(5.83)^2

    After computing I get 1.27*10^5 J --> Can anyone verify?

    b) This part is where I get lost... The change in linear momentum is the same thing as impulse. Impulse is m(vf-vi) --> I know I cannot just subtract the velocities that are given because they are not in same direction. They have direction and magnitude. I must use vectors to find the vf-vi.

    In vector notation, vf-vi is that same things as saying vf+(-vi)

    Turns out that the initial velocity only has a y component, which makes its velocity (after computing with the vectors) = 5.83 m/s

    The final velocity only has an x component and a 0 y component, which makes its velocity ( after computing with a y and x component) = -7.22 m/s

    After getting this far, if I did it right, I assume that now I must do another vector problem with the vector final + (- vector initial)

    I draw the final vector in the negative direction (west) and lable it with -7.22, then I flip the direction of the north vector (v initial) making it south to subtract from the v final. I get a vector pointing west, plus a vector pointing south. I draw an arrow from the tail of first to tip of last.

    I then have a right angle, do pythagorean theroem to find

    86.1173 = resultant^2 ... I take the square root to get 9.27

    Is this value vf-vi?

    If so, I multiply this by the mass of the truck to get the change in momentum?

    So..... (9.27) * 14000 kg = 1.30*10 ^5 kg*m/s

    To get the direction, I do the arctan of what values? Is it the velocity of final/intial??

    Thanks for the help!
     

    Attached Files:

  2. jcsd
  3. Aug 7, 2017 #2

    BvU

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    Yes. If you make a skeetch you will also see which ##\arctan## you need for the angle
     
  4. Aug 7, 2017 #3
    Okay so I would use the values though for the velocity? Because momentum is in the same direction as the velocity right?

    So it will be the arc tan of (initial velocity / the final velocity) (I drew out the triangle and tan is the opposite over adjacent)

    The opposite is the velocity initial and the adjacent is the velocity final.

    So theta = arctan(5.833/-7.22) ?

    Does this seem correct?
     
  5. Aug 7, 2017 #4

    BvU

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    Yes

    Note:
    but you meant -7.22 m/s East (but you processed correctly).

    No. Can you show the drawing ? In which quadrant does the vector ##\Delta\vec v## end up ?
    And: in which quadrants do arctan results end up ?
     
  6. Aug 7, 2017 #5
    Isn't it drawn like this?

    The resultant is in quadrant three?
     

    Attached Files:

  7. Aug 7, 2017 #6
    Oh wait.. the downwards line has to be negative as well?
    So it is arc tan (-5.8333/-7.22)

    Is that better now?
     
  8. Aug 7, 2017 #7
    Thus it is 38.9 degrees below negative x axis.
     
  9. Aug 7, 2017 #8

    haruspex

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    Yes.
     
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