After driving 20.0 km west, a car turns southwest and proceeds another 50.0 km. What is the displacement (in kilometers) of the car from the point of origin (magnitude only)?
find the N,S, E, W components
W: 20km+50cos45
S: 50sin45
now, magnitude= sqrt( W^2 + S^2)
S=1/2gt^2
To find the displacement of the car from the point of origin, we can use vector addition. We need to determine the x and y components of the displacement.
First, let's consider the 20.0 km westward displacement. Since this is a straight line purely in the x-axis direction, the x-component of this displacement would be -20.0 km and the y-component would be 0 km.
Next, let's consider the 50.0 km southwestward displacement. To calculate the x-component of this displacement, we need to find the horizontal component of the southwest vector. Since the southwest direction is between the west and the south direction, we need to find the proportion of the southwest vector in each direction.
The cosine of the angle between the southwest vector and the x-axis will give us the proportion of the southwest vector in the x-direction. The sine of the angle will give us the proportion of the southwest vector in the y-direction.
Let's calculate the x-component of the southwest vector:
cos(45°) = adjacent/hypotenuse
x-component = adjacent = cos(45°) * 50 km = 35.355 km (rounded to three decimal places)
Now, let's calculate the y-component of the southwest vector:
sin(45°) = opposite/hypotenuse
y-component = opposite = sin(45°) * 50 km = 35.355 km (rounded to three decimal places)
To find the total displacement, we need to add the x-components and y-components separately:
Total x-component = -20.0 km + 35.355 km = 15.355 km (rounded to three decimal places)
Total y-component = 0 km + 35.355 km = 35.355 km (rounded to three decimal places)
Finally, we can use the Pythagorean theorem to find the magnitude of the displacement:
Displacement (magnitude) = sqrt((Total x-component)^2 + (Total y-component)^2)
Displacement (magnitude) = sqrt((15.355 km)^2 + (35.355 km)^2)
Displacement (magnitude) = sqrt(235 + 1250) km
Displacement (magnitude) = sqrt(1485) km
Displacement (magnitude) ≈ 38.547 km (rounded to three decimal places)
Therefore, the displacement (magnitude only) of the car from the point of origin is approximately 38.547 km.