A teacher sends her students on a treasure hunt.She gives the following instructions: 1.Walk 300 m north.
2)Walk 400 m northwest.
3)Walk 700 m east-southeast and the treasure is buried there.
As all the other students walk off following the instructions,Jane physics student quickly adds the displacements and walks in a straight line to find the treasure.How far and in what direction does Jane need to walk?

Question

Quizplus · Accepted Answer

To find the total displacement of the treasure from the starting point, we need to add the three given displacements.

The first displacement of 300 m north is solely in the y-direction.

For the second displacement of 400 m northwest, we can break it down into its x and y components using the angle of 315 degrees (45 degrees from west):

x = 400 cos(315) = -282.8 m (east is positive, so west is negative)
y = 400 sin(315) = 282.8 m (north is positive)

Note the negative sign on the x-component since we are going west.

For the third displacement of 700 m east-southeast, we need to use the angle of 67.5 degrees from east-south-east:

x = 700 cos(67.5) = 309.8 m (east is positive)
y = -700 sin(67.5) = -615.9 m (north is positive)

Note the negative sign on the y-component since we are going south.

Now we can add up the x and y components:

Total x displacement = 300 - 282.8 + 309.8 = 327.0 m (east is positive) 
Total y displacement = 0 + 282.8 - 615.9 = -333.1 m (north is positive)

The distance Jane needs to walk to get to the treasure is the magnitude of the total displacement:

distance = sqrt((327.0)^2 + (-333.1)^2) = 481.1 m

The direction Jane needs to walk can be found using the inverse tangent function:

tan(theta) = (-333.1) / 327.0 
theta = tan^-1(-333.1 / 327.0) = -40.9 degrees

Note that we have a negative angle because we are going west of north. However, the question is asking for the direction relative to east, so we need to add 90 degrees to this angle:

theta = -40.9 + 90 = 49.1 degrees

Finally, we need to take the complement of this angle (90 - theta) to get the angle relative to north of east:

angle = 90 - 49.1 = 40.9 degrees

So Jane needs to walk 481 m in a direction 40.9 degrees north of east to find the treasure.

A Teacher Sends Her Students on a Treasure Hunt