A man in search of his dog drives first 10 mi northeast, then 12 mi straight south, and finally 8 mi in a direction 30 degree north of west . What are the magnitude and direction of his resultant displacement?
All angles are measured CCW from +x-axis.
D = 10[45o] - 12i + 8[150o].
D = 7.07 + 7.07i - 12i - 6.93 + 4i = 0.14 - 0.93i = 0.94mi[-81.4] = 0.94mi[278.6o].
thanks a lot
sir.
To find the magnitude and direction of the resultant displacement, we can break down the three displacements into their respective x and y components.
1. The first displacement of 10 miles northeast can be broken down into its x and y components. Since it is going northeast, the displacement has equal amounts of north and east components. To find these components, we can use trigonometry. We know that the angle between the displacement vector and the positive x-axis is 45 degrees (since northeast is halfway between north and east). Therefore, the x-component is 10 miles * cos(45 degrees) = 10 miles * √2/2 = 5√2 miles, and the y-component is 10 miles * sin(45 degrees) = 10 miles * √2/2 = 5√2 miles.
2. The second displacement of 12 miles straight south only has a negative y-component since it is going straight south. Therefore, the x-component is 0 miles, and the y-component is -12 miles.
3. The third displacement of 8 miles in a direction 30 degrees north of west can also be broken down into its x and y components using trigonometry. The angle between the displacement vector and the positive x-axis is 30 degrees west of north, which is equivalent to 60 degrees east of north. Therefore, the x-component is 8 miles * cos(60 degrees) = 8 miles * 1/2 = 4 miles, and the y-component is -8 miles * sin(60 degrees) = -8 miles * √3/2 = -4√3 miles.
Now, we can add up the x-components and the y-components separately to find the resultant displacement:
x-component = 5√2 miles + 0 miles + 4 miles = 9√2 miles
y-component = 5√2 miles - 12 miles - 4√3 miles = 5√2 - 4√3 - 12 miles
To find the magnitude of the resultant displacement, we can use the Pythagorean theorem:
magnitude = √((x-component)^2 + (y-component)^2)
magnitude = √((9√2 miles)^2 + (5√2 - 4√3 - 12 miles)^2)
Once we calculate the magnitude, we can find the direction using the inverse tangent function:
direction = tan^(-1)(y-component / x-component)
Plugging in the values, we can calculate the magnitude and direction of the resultant displacement.