A truck driver travels a distance of 100 miles due East. He then turns at an angle of 35 degrees S of E and travels a distance of 50 miles.

What is the resultant displacement?
What is the direction?

To find the resultant displacement, we can break it down into its horizontal and vertical components using trigonometry.

Horizontal component:
The truck initially travels due East, which means its horizontal displacement is 100 miles.

Vertical component:
The truck turns at an angle of 35 degrees S of E and travels a distance of 50 miles. To find the vertical displacement, we need to find the component of the distance traveled in the southward direction, which is the opposite side of the angle.

Opposite side = 50 miles * sin(35 degrees)

Now that we have the horizontal and vertical components, we can find the resultant displacement using the Pythagorean theorem:

Resultant displacement = √(Horizontal component^2 + Vertical component^2)

Let's calculate it:

Horizontal component = 100 miles
Vertical component = 50 miles * sin(35 degrees) ≈ 28.70 miles

Resultant displacement = √(100^2 + 28.70^2) ≈ 105.33 miles

So, the resultant displacement is approximately 105.33 miles.

To find the direction, we can use trigonometry again. The angle at which the resultant displacement is measured can be found using the inverse tangent or arctangent function:

Direction = arctan(Vertical component / Horizontal component)

Let's calculate it:

Direction = arctan(28.70 miles / 100 miles) ≈ 16.71 degrees

Therefore, the direction of the resultant displacement is approximately 16.71 degrees S of E.

Just add the two vectors.

The eastward displacement is
X = 100 + 50 cos35 = 140.96 miles

The northern displacement (assuming north is positive) is
Y = -50 sin35 = -28.68 miles
(The minus sign is there because the second vector is south of east, not north).

Total displacementis
= sqrt[(140.96)^2 + (-28.68)^2]

The direction(measured south of east) is tan^-1 (28.68/140.96) = 11.5 degrees