a car travels 215 km w and then 85 km south west. what is the magnitude and direction?

Answer 1

You can draw a diagram to work out the total displacement and bearing.

OK, this is very hard to explain without a sheet of paper, but you can calcuate it. The car moves 215 km due West. Then you have to imagine the 85 km Southwest as X km West and X km South. If it's exactly Southwest, it will travel the same distance west than south throughout those 85 km.

I would use Pythagoras's theorem to work it out. a^2+b^2=c^2. a is the distance it travels south and b is the distance it travels west. c is 85 km southwest. Since a is equal to b, and we know c, let's change the formula to 2a^2=85^2

Therefore,
2a^2=7225
a^2=3612.5
a=60.104076

So the car, travelling 215 km west and 85 km southwest is the same as it travelling 215 km west, 60.1 km south and 60.1 km west again. Or, 275.1 km west and 60.1 km south.

Use Pythagoras again to work out the total displacement. a^2+b^2=c^2 so therefore 275.1^2+60.1^2=c^2
79294.75=c^2
c=281.59324 km

For the direction, you need to use trigonometry. If you draw a right-angled triangle with one side 275.1 km, one side 60.1 km (the last side is 281.6 km as we just worked out). When you draw it, the right angle will be to the bottom right of the paper. 275.1 km is to the BOTTOM of the paper (moving west) and 60.1 km is to the RIGHT of the paper (moving south). You want to work out the angle on the TOP-RIGHT of the triangle. You know the opposite and adjacent side, so I use the tan rule.

tanx=opp/adj.
tanx=275.1/60.1
x=77.676
You need to add 180 degrees to get the bearing, from the diagram. The bearing is about 257.7 degrees, a WSW direction but slightly more to the W than straight, due WSW.

Answer 2

It's easiest to do this problem using the unit vectors i = <1, 0> and j = <0, 1>.

Let u be the first distance travelled and v the second distance travelled.

u = -215 i
v = (-85 cos 45) i + (-85 sin 45) j = -60.1 i - 60.1 j

u + v = -275.1 i - 60.1 j

The magnitude is sqrt[x^2 + y^2]

= sqrt[ 275.1^2 + 60.1^2] = 281.6 km

The direction is found using theta = arctan(y / x)

theta = arctan( -60.1 / -275.1) = 12.3 degrees


The resultant vector is 281.6 km at 12.3 degrees S of W.

Answer 3

My friend, please complete your question. .."What is the magnitude and direction of......................?"