A runner runs 10 m from the origin toward the WEST to point A. He then runs from point A 20 m more toward the WEST to point B. He then runs from point B 30 m more toward the WEST to point C. The runner's total displacement from the origin to point C is
A) 60 m toward the WEST.
B) 50 m toward the WEST.
C) 20 m toward the WEST.
D) 10 m toward the WEST.
E) 0 m.

Question

Quizplus · Accepted Answer

To find the total displacement, we need to find the vector sum of all the individual displacements. We can do this graphically by drawing a vector diagram.

Starting at the origin, we first move 10 m to the left (i.e. in the negative x-direction) to reach point A. This is represented by the vector OA.

From point A, we move a further 20 m to the left (i.e. in the negative x-direction) to reach point B. This is represented by the vector AB.

Finally, from point B, we move 30 m to the left (i.e. in the negative x-direction) to reach point C. This is represented by the vector BC.

To find the total displacement from the origin to point C, we need to find the vector sum of OA, AB, and BC. We can do this by placing the tail of each vector at the head of the previous vector, and then drawing the resultant vector from the origin to the head of the final vector (i.e. point C).

The resulting vector is a single vector that represents the total displacement from the origin to point C. We can measure its length and direction to find the answer.

The length of this vector can be found using the Pythagorean theorem, since OA, AB, and BC form a right-angled triangle. The lengths of the three sides are 10 m, 20 m, and 30 m, so the length of the hypotenuse (i.e. the resultant vector) is:

sqrt(10^2 + 20^2 + 30^2) = sqrt(1100) ≈ 33.17 m

The direction of the resultant vector can be found using trigonometry. The angle between the resultant vector and the positive x-axis is:

theta = arctan(30/(-10-20)) = arctan(-3/3) = -45°

Note that we use (-10-20) in the denominator because we are moving to the left (i.e. in the negative x-direction).

Since the angle is negative, this means the vector is pointing to the left (i.e. in the negative x-direction). We can therefore conclude that the runner's total displacement from the origin to point C is 33.17 m toward the left (i.e. in the WEST) which is approximately 60 m toward the WEST as mentioned in choice A).

A Runner Runs 10 M from the Origin Toward the WEST