A car is driven east for a distance of 42 km, then north for 25 km, and then in a direction 31° east of north for 26 km. Determine (a) the magnitude (in km) of the car's total displacement from its starting point and (b) the angle (from east) of the car's total displacement measured from its starting direction.
To determine the car's total displacement, we need to break down each leg of the journey into its horizontal and vertical components.
First, let's calculate the horizontal and vertical components of the car's displacement for each leg.
For the eastward leg:
- Horizontal component: 42 km (since the car is driving purely east)
- Vertical component: 0 km (since there is no northward or southward movement)
For the northward leg:
- Horizontal component: 0 km (since there is no eastward or westward movement)
- Vertical component: 25 km (since the car is driving purely north)
For the leg 31° east of north:
We need to calculate the horizontal and vertical components using trigonometry.
- Horizontal component: 26 km * cos(31°) (the adjacent side of the right triangle formed by the displacement)
- Vertical component: 26 km * sin(31°) (the opposite side of the right triangle formed by the displacement)
Now, let's calculate the horizontal and vertical components for the leg 31° east of north:
- Horizontal component: 26 km * cos(31°) ≈ 22.293 km
- Vertical component: 26 km * sin(31°) ≈ 13.312 km
Next, we add up the horizontal and vertical components to find the total horizontal and vertical displacements.
Total horizontal displacement: 42 km + 0 km + 22.293 km ≈ 64.293 km
Total vertical displacement: 0 km + 25 km + 13.312 km ≈ 38.312 km
Finally, we can calculate the magnitude and direction of the car's total displacement.
(a) Magnitude of total displacement:
The magnitude can be found using the Pythagorean theorem, which states that the magnitude (M) of the displacement is given by the square root of the sum of the squares of the horizontal (h) and vertical (v) components.
M = √(h^2 + v^2)
M = √(64.293^2 + 38.312^2)
M ≈ √(4137.15 + 1467.69)
M ≈ √(5604.84)
M ≈ 74.9 km
Therefore, the magnitude of the car's total displacement is approximately 74.9 km.
(b) Angle of total displacement measured from its starting direction:
The angle can be found using trigonometry. We use the inverse tangent function (tan^-1) to find the angle from the horizontal axis (east).
Angle = tan^-1(v/h)
Angle = tan^-1(38.312/64.293)
Angle ≈ 31.6°
Therefore, the angle of the car's total displacement measured from its starting direction (east) is approximately 31.6°.