A car is driven 215 km west and then 85 km southwest. What is the displacement of the car from the point of origin magnitude and direction draw a diagram.?
I kinda understand how to do this but in order to solve this you need to get a 45 degree angle and i don't know where that comes from
To find the displacement of the car, you can break down the distances traveled into their respective horizontal (west/east) and vertical (north/south) components.
For the first part, when the car is driven 215 km west, the horizontal component is -215 km (since west is negative on a Cartesian coordinate system), and the vertical component is 0 km (as there is no movement in the north/south direction).
For the second part, when the car is driven 85 km southwest, you need to determine the horizontal and vertical components of this displacement. The southwest direction can be represented as a combination of west and south directions. Since the car moved southwest, it means it traveled 85 km in a direction 45 degrees below the horizontal line.
To find the horizontal and vertical components of 85 km southwest, you need to use trigonometry. By drawing a right-angled triangle, you can apply trigonometric functions such as sine and cosine. In this case, you can use cosine to find the horizontal component and sine to find the vertical component, since the angle is measured with respect to the horizontal axis.
By applying the cosine function, you can determine that the horizontal component of the displacement is -85 km multiplied by cos(45°), which is approximately -60.14 km.
Similarly, by applying the sine function, you can determine that the vertical component of the displacement is -85 km multiplied by sin(45°), which is approximately -60.14 km.
To find the total displacement, you need to sum up the horizontal and vertical components obtained from both parts of the journey:
Horizontal displacement = -215 km + (-60.14 km) ≈ -275.14 km
Vertical displacement = 0 km + (-60.14 km) ≈ -60.14 km
The magnitude of the displacement can be found using the Pythagorean theorem, which states that the square of the hypotenuse (in this case, the displacement) is equal to the sum of the squares of the other two sides.
Magnitude of the displacement = √((-275.14 km)² + (-60.14 km)²) ≈ 284.32 km
The direction of the displacement can be determined by calculating the angle between the displacement vector and the positive x-axis of the Cartesian coordinate system. This can be done using trigonometry again. In this case, you can use the tangent function:
Direction = arctan((-60.14 km)/(-275.14 km)) ≈ 11.83°
To summarize, the displacement of the car from the point of origin is approximately 284.32 km in a direction of 11.83 degrees below the positive x-axis (west).
To solve the problem, we can break down the displacement into two components: one along the west direction and the other along the southwest direction.
1. First, let's find the displacement in the west direction. The car is driven 215 km west, so the magnitude of this displacement is 215 km towards the west.
Sketch:
----> (215 km west)
2. Next, let's find the displacement in the southwest direction. The car is driven 85 km southwest, which we can break down into two perpendicular components: one towards the south and the other towards the west.
Sketch:
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| (85 km southwest)
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3. Now, we need to find the magnitudes of the south and west components of the southwest displacement. Since the southwest angle is 45 degrees, we can use trigonometry to find these magnitudes. Given that the hypotenuse (85 km southwest displacement) and one angle (45 degrees) are known in a right-angled triangle, we can find the magnitudes of the sides using the trigonometric functions.
magnitude of south component = 85 km * sin(45 degrees)
magnitude of west component = 85 km * cos(45 degrees)
magnitude of south component = 85 km * 0.707 ≈ 60.5 km
magnitude of west component = 85 km * 0.707 ≈ 60.5 km
4. Now, we can add the displacements in the west and southwest directions to find the total displacement.
Total displacement in the west direction = 215 km
Total displacement in the south direction = 60.5 km
By adding these two displacements, we can use the Pythagorean theorem to find the magnitude of the total displacement.
Magnitude of the total displacement = sqrt((215 km)^2 + (60.5 km)^2)
Magnitude of the total displacement ≈ sqrt(46225 km^2 + 3660.25 km^2)
≈ sqrt(49885.25 km^2)
≈ 223.23 km (rounded to two decimal places)
The magnitude of the total displacement is approximately 223.23 km.
5. Finally, let's determine the direction of the total displacement. We can use trigonometry to find the angle between the total displacement vector and the west direction.
Angle = atan((magnitude of south component) / (magnitude of west component))
Angle = atan(60.5 km / 215 km)
Angle ≈ atan(0.281)
≈ 15.75 degrees (rounded to two decimal places)
Thus, the direction of the displacement is approximately 15.75 degrees north of west.
To summarize, the displacement of the car from the point of origin is approximately 223.23 km at an angle of 15.75 degrees north of west.