Vectors and Projectiles - Mission VP4 Detailed Help

In the Vector Addition Lab, the following data was collected for determining the displacement from the door of the Physics classroom to another location in the building: 2 m, West; 14 m, South; 22 m, East; 19 m, North; and 2 m, West. The direction of the resultant displacement is closest to ___ degrees. 
 
(Note: Numbers are randomized numbers and likely different from the numbers listed here.)


 
There are a several conventions for expressing the direction of a vector. The convention used here is the counterclockwise (CCW) from East convention (see Dictionary section). It is possible that a student could do all his/her math correctly but miss the question because he/she failed to use the CCW convention.


 
An effective strategy for all questions in this mission will center around a rough sketch of the addition of two vectors (See Think About It section). Consider the following steps:
  • Begin by simplifying the collection of small displacements by adding all the east-west vectors together. Consider west to be the negative direction and add the negative westward displacements to the positive eastward displacements. Record the result and label as ∑E-W (for sum of the east and west vectors).
  • Repeat the process of adding the north-south vectors together, considering south to be the negative direction. Record the result and label as ∑N-S (for sum of the north and south vectors). The two vector sums will now be added together.
  • Sketch the first vector (the ∑E-W vector) in the appropriate direction. Place an arrowhead at the end of the vector and label the vector's magnitude on the sketch.
  • Starting at the arrowhead of the first vector, draw the second vector (the ∑N-S vector) in the appropriate direction and to the approximate length. Put an arrowhead at the end of the vector and label the vector's magnitude on the sketch.
  • Draw the resultant vector from the tail of the first to the arrowhead of the second vector. Label the vector as R(for resultant) and put an arrowhead at the end of the resultant vector.
  • At the tail of the resultant, label the angle between the resultant vector and the adjacent leg as the angle theta (Θ).
  • Since the two vectors being added and the resultant form a right triangle, SOH CAH TOA can be used to calculate the angle Θ (see Math Magic section). The tangent function can be used to relate the angle to the lengths of the horizontal and vertical legs of this right triangle.
  • The angle Θ is the angle between the resultant Rand the adjacent leg. Thus,Θ is NOT necessarily the answer to the question since you are to state the direction using the counterclockwise from East convention (see the Define Helpsection). Since Θ is the angle between the resultant and one of the axes, the angle between the resultant and east as measured counterclockwise can be determined using the value of Θ.


 
The head-to-tail method of vector addition should be used to create a rough sketch of this physical situation. The sum of the east-west vectors and the north-south vectors can be added together. In this method, the first vector (the ∑E-W vector) is sketched (not to scale) in its indicated direction. The second vector (the ∑N-S vector) is then sketched (not to scale) starting at the head (arrowhead) of the first vector. The resultant vector is then drawn from the tail of the first vector to the head of the last vector. A trigonometric function can then be used to calculate the magnitude of the resultant (see Math Magic section).


 
SOH CAH TOA
The trigonometric functions sine, cosine and tangent can be used to express the relationship between the angle of a right triangle and the lengths of the adjacent side, opposite side and hypotenuse. The meaning of the three functions are:
 
sine Θ = (length of opposite side / length of hypotenuse)
cosine Θ = (length of adjacent side / length of hypotenuse)
tangent Θ = (length of opposite side / length of adjacent side)


 
Counterclockwise from East Convention for Vector Direction
The direction of a vector is often expressed using the counterclockwise (CCW) convention. According to this convention, the direction of a vector is the number of degrees of rotation which the vector makes counterclockwise from East.