One Chapter 3

• Chapter two dealt with motion on a line. Very few objects move this way.
  • To go to the local store, you no doubt have to travel around many corners.
  • Airplanes must gain and lose elevation.
  • The International Space Station travels in a nearly circular orbit.
• We must consider the vector nature of quantities when discussing more than one dimensional motion.

• Displacement - The length and direction of the straight line joining an object's initial and final positions.
  • The path taken between initial and final positions is not important.
    • If you want to go from home to the University, it does not matter if you take Deerfoot Trail to Memorial Drive to Crowchild Trail to University Drive, OR 14th Street to Glenmore Trail to Crowchild Trail to University Drive. As long as you get there.
  • Objects which start at the same point and end at the same point may travel different distances, but will have the same displacement.
• Methods to determine displacement in more than one dimension:
  • Scale diagram
  • Trigonometry
• Trigonometry is by far the more valuable method to use to determine displacement.

• Velocity - The rate of change of displacement with time.
  • Direction is important.
• Speed - The rate of change of distance with time.
  • Direction is not important.
• Average speed and velocity may be quite different.
• Instantaneous speed and velocity are always numerically equal.
• An example of velocity is 20 m/s [W 20^o N]. The notation [W 20^o N] tells us that to draw the vector in a scale diagram, the baseline of the protractor must point to the west (the first direction listed), and we turn 20^o to the north (the second direction listed). An arrow is then drawn of appropriate length to represent the vector. To draw an arrow of appropriate length requires a scale such as 1 cm = 5 m/s. Therefore the arrow would be 4 cm long.

• Vectors that fall on the same line can just be mathematically added.
• To find the vector sum of the vectors +5 m/s and -3 m/s just add the numbers +5 m/s + (-3 m/s) = +2 m/s. Therefore, the answer is +2 m/s.
• Vectors not on the same line must be added by using:
  • Scale diagram
  • Trigonometry
• Rule for Adding Vectors - The vectors to be added are joined in any order Tip-to-Tail, and the vector representing their sum joins the tail of the first vector, pointing towards the tip of the last vector added.
  • The order of addition is not important.
  • Vector addition is commutative.
• For the example above, (+5m/s + (-3 m/s)), the scale diagram would look like the following.

• Consider the addition of three displacements. A woman walks 300 m [E], then 200 m [W 20^oS] and finally 150 m [N 40^o E]. What is her displacement after she finishes?
• We will answer by using a scale diagram. Below you will find the steps to follow.
• Choose a scale for your diagram.

• Choose a reference point (reference frame) from which to start.

• Draw an arrow to represent any one of the vectors with the initial point (tail) of the arrow at the reference point. The arrow should point in the correct direction with respect to your reference point and be of proper length according to your scale.

• Draw another vector in a similar way, only place the tail of this arrow at the tip of the last one drawn (Tip-to-Tail).

Repeat the previous step until all of the vectors are drawn.

• Draw in the Resultant Vector by drawing an arrow from the tail of the first vector drawn, pointing towards the tip of the last vector drawn.

• Determine the magnitude of the resultant vector by measuring its length and using the scale you chose.

• Determine the direction of the resultant vector by measuring angles from the reference point (reference frame) you chose.

• State your answer.

• To subtract vectors, add the negative of the one to be subtracted. The negative of a vector is simply a vector of the same magnitude, only pointing in the opposite direction.

• An alternative method is:
  • draw the vectors tail-to-tail
• The resultant is from the tip of the vector being subtracted towards the tip of the vector it is being subtracted from. For the above example that would be as follows

• Unlike addition of vectors, subtraction of vectors is not commutative. That is to say,

• Vector subtraction can be determined by:
  • Scale diagram
  • trigonometry
• To review vector addition with a scale diagram, click Here and view or download the file called Vector Addition1.pps.

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October 29, 2013