What is the total displacement of a delivery truck given its route and time?

[JackandJones]

A delivery truck drives 14.0 km [E], 6.0 km [N], 12.0 km [ N 15° E], and then 2.0 km [N 65° E]. This takes him 42 minutes. Calculate the truck's distance and displacement.


When you draw this diagram you basically get two triangles. One large triangle and one smaller triangle. You need to find displacement. For the large triangle you can use pythagorean theorem, for the small triangle---that is where I am having problems.

I split 12 km (N 15 degrees E) into components: so I got 12 x sin (15) and 12 x cos (15) for the one small triangle

I split 2 km (N 65 degrees E) into components so I got 2 x sin (65), and 2 x cos (65)

So the total for x = 4.92 km and y = 12. 45 km...

Now what do I do?? I was thinking Cosline Law: but I don't know what to sub in for Cos.

Any help is appreciated!

 

[PhanthomJay]

Try splitting it into three triangles...one big one (the first one with the 14 and 6 km legs, and the next one with the 12 km hypotenuse, and the last one with the 2 km hyphyotenuse. Then add up the lengths to get the distance traveled, and and use pythagorus and trig on the giant triangle (the one formed with the hypotenuse running from the start point to the end point) to get the displacemnt magnitude and direction. Be sure to draw a sketch!

 

[JackandJones]

Okay I tried doing that...but I am not sure it is correct. Could you show me some calculations of what you got? And/or show me a diagram?!

I appreciate the help!

 

[PhanthomJay]

I didnt carefully check your trig, but you already have found the x and y components of each of the last 2 vectors, and you can find the x and y components of the first vector in the same manner..to get the total displacement, that's the square root of the sum of the squares of the x and y components, R = sqrt [(sum of x)^2 + (sum of y)^2]. See here:
http://hyperphysics.phy-astr.gsu.edu/hbase/vect.html
To get the distance traveled, add up the actual lengths of travel.

 

[JackandJones]

I got 25 km N 65 degrees E...that can't be right, can u try and calculate the final answer?

 

[PhanthomJay]

You found the sum of the x components of the last 2 vectors, which is 4.92. Now add it to the x component of the first triangle, which is 14, to get the total resultant displacement in the x direction, R_x.

You have also found the y components of the last 2 vectors, which is 12.45. Now add it to the y component of the first triangle, which is 6, to get the total resultant displacement in the y direction, R_y.

Now use pythagorus to find R, and trig to find theta.