Mariana finds a cave to explore. Starting at the cave entrance, Mariana first fo

Question

Mariana finds a cave to explore. Starting at the cave entrance, Mariana first follows a passage 75.0 m north, then turns and moves 250 m east, then goes 124 m at an angle 30.0° north of east, and finally moves 172 m south. Find the resultant displacement from the cave entrance. Shown is a sketch of the situation not drawn to scale. (Give the magnitude of the displacement in m and the direction in degrees south of east.) Cave entrance Final position G What is the resultant displacement from the entrance? magnitude direction 90 Remember that the angle is found by taking the inverse tangent of the y-component of the total displacement divided by the x-component. Be sure to include signs in your calculation.°south of east

Explanation / Answer

1.

Suppose given that Vector is R and it makes angle A with +x-axis, then it's components are given by:

Rx = R*cos A

Ry = R*sin A

Using above rule:

R1 = 75 m in North,

R1x = 0 m

R1y = 75 m

R2 = 250 m in East

R2x = 250 m

R2y = 0 m

R3 = 124 m & angle = 30 deg above +ve x-axis

R3x = R3*cos A3 = 124*cos 30 deg

R3x = 107.39 m

R3y = R3*sin A3 = 124*sin 30 deg

R3y = 62 m

R4 = 172 m in South

R4x = 0 m

R4y = -172 m

Now, Net displacement will be

Rnet = Rnet)x + Rnet)y

Rnet = (R1x + R2x + R3x + R4x) i + (R1y + R2y + R3y + R4y) j

Using above values

Rnet = (0 + 250 + 107.39 + 0) i + (75 + 0 + 62 - 172) j

Rnet = 357.39 i - 35 j

|Rnet| = sqrt (Rnet_x^2 + Rnet_y^2)

|Rnet| = sqrt (357.39^2 + (-35)^2)

|Rnet| = 359.099 m

|Rnet| = 359.1 m

Direction of Rnet will be

Direction = arctan (Rnet_y/Rnet_x)

Direction = arctan (35/357.39) = 5.59 deg

Direction = 5.59 deg below positive x-axis = 5.59 deg South of East

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