# determine the resultant speed of the plane

a plane flying due east at 200 km/h encounters a 40-km/h wind blowing in the north-east direction. the resultant velocity of the plane is the vector sum v = $v_1$ + $v_2$, where $v_1$ is the velocity vector of the plane and $v_2$ is the velocity vector of the wind. the angle between $v_1$ and $v_2$ is pi/4. determine the resultant speed of the plane(the length of the vector v).

how do you approach this problem? Im not sure why the angle comes into play.

is the length of the vector just the magnitude?

tried that and then doing side angle side to solve for the length.so i did a2=2002+402-2(200)(40)(cos135). But then I ended up getting 201 for the answer but it should be 230. Where did I do wrong?

You have to use vectorial addition - the angle is the difference of direction. Hint: Draw a triangle with

• a point (e.g. A) as the current position of the plane
• one side is for the velocity of the plane (going east, that is a line/vector going horizontally to the right from a, length: something representing 200, e.g. 5inch/cm)
• one side is for the velocity of the wind (going north east, length representing 40km-h, e.g. 1 inch/cm). This line has to be appended at the end of your first line/vector
• then draw a line from the starting point A to the end of the second line/vector
• that vector represents the resulting velocity vector
• in order to calculate it length you could e.g. use trigonometry

By the way, the information about the angle (pi/4) can be retrieved from the directions (east and north east) already and is thus redundant. If you draw your triangle you will see that the angle does have a big influence on the length of the resulting velocity vector.

• @alexThefourth: please comment whether this helped you or mark as an answer Jan 13, 2014 at 8:50
• I tried that and then doing side angle side to solve for the length.so i did $a^2$=$200^2$+$40^2$-2(200)(40)(cos135). But then I ended up getting 201 for the answer but it should be 230. Where did I do wrong? Jan 13, 2014 at 15:51
• You didn't go wrong - just your result is wrong. Make sure your calculator is in degree mode Jan 13, 2014 at 16:01
• I edited my answer - some items weren't exact Jan 13, 2014 at 16:03
• I calculated, too. The solution of your equation is indeed 230,... . However, the solution should (in my opinion) be somewhat about 174,03 because (see dot 3) you have to append the second line at the end of the first line. This would change the angle from 135 to 45 degrees. Jan 13, 2014 at 16:09