A car travels 3.0km due west and then 2.0km to the north. Find the magnitude and
direction of the car’s resultant displacement?
X = -3.0km
Y = 2km
Tan Ar = Y/X = 2/-3 = -0.66667
Ar = -33.69o = Reference angle.
A = -33.69 + 180 = 146.3o, CCW.
D=X/cos A = -3/cos146.3=3.61km[146.3o]
To find the magnitude and direction of the car's resultant displacement, we can use the Pythagorean theorem and trigonometric functions.
Step 1: Draw a diagram to visualize the car's movements.
Step 2: Use the Pythagorean theorem to find the magnitude (or length) of the car's resultant displacement. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the northward movement can be represented as the vertical side and the westward movement as the horizontal side of a right triangle. Using the formula:
c² = a² + b²
where c is the hypotenuse (resultant displacement), a is the westward movement, and b is the northward movement.
Substituting the values we have:
c² = (3.0 km)² + (2.0 km)²
c² = 9.0 km² + 4.0 km²
c² = 13.0 km²
Taking the square root of both sides:
c = √13.0 km
c ≈ 3.61 km (rounded to two decimal places)
Therefore, the magnitude of the car's resultant displacement is approximately 3.61 km.
Step 3: Use trigonometric functions to find the direction of the car's resultant displacement. We can use the tangent function to determine the angle in relation to a reference direction (usually north).
The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle. In this case, the opposite side is the northward movement (2.0 km) and the adjacent side is the westward movement (3.0 km).
Using the formula:
tanθ = opposite/adjacent
tanθ = (2.0 km)/(3.0 km)
tanθ ≈ 0.6667
To find the angle θ, we can take the inverse tangent (or arctan) of 0.6667 using a calculator.
θ ≈ arctan(0.6667)
θ ≈ 33.69 degrees
Therefore, the direction of the car's resultant displacement is approximately 33.69 degrees north of west.