An automobile leaves the store, first going south at 88 km/h for 2 hours, and then turns on a road heading 20 degrees east of north at 55 km/h for 1.5 hours. Calculate the (x,y) components of the net displacement of the automobile from the starting position, where the x-axis is east and y-axis is north. Calculate the magnitude and direction of the automobile's net displacement. Give the magnitude in km and its direction using points of the compass.
D1 = 88 * 2 = 176km, S.
D2 = 55 * 1.5 = 82.5km[70o] N. of E.
X = 82.5*Cos70 = 28.22 km.
Y = -176 + 82.5*sin70 = -98.5 km.
D = sqrt(X^2+Y^2) = sqrt(28.22^2+98.5^2) = 102.5 km.
Tan A = Y/X = (-98.5)/28.22 = -3.49043, A = -74o = 74o S. of E.
= Direction.
To calculate the (x,y) components of the net displacement, we need to break down the motion into its horizontal (x) and vertical (y) components.
First, let's calculate the horizontal component:
- The car traveled south, so there is no horizontal displacement during this part.
Next, let's calculate the vertical component:
- The car traveled south at a speed of 88 km/h for 2 hours, so the vertical displacement is given by:
Vertical displacement = Speed * Time = 88 km/h * 2 hours = 176 km south
To find the (x, y) components, we need to find the components of the vertical displacement. We can use trigonometry to split the displacement into its horizontal and vertical components.
- The car turned on a road heading 20 degrees east of north, which means it's essentially traveling 70 degrees north of east (90 - 20).
- The vertical component (y) of the displacement is given by the vertical displacement multiplied by the sine of the angle:
y = vertical displacement * sin(angle) = 176 km * sin(70°)
- The horizontal component (x) of the displacement is given by the vertical displacement multiplied by the cosine of the angle:
x = vertical displacement * cos(angle) = 176 km * cos(70°)
Now, let's calculate the magnitude of the net displacement:
- The magnitude of the net displacement is given by the Pythagorean theorem:
Magnitude = sqrt(x^2 + y^2)
Finally, let's calculate the direction of the net displacement using the compass points:
- We can use inverse tangent (arctan) to find the angle between the x-axis (east) and the net displacement vector. The result will be in radians.
- Convert the result to degrees and adjust for the range of 0 to 360 degrees, taking into account the appropriate quadrant.
So, to summarize:
- The x-component (horizontal) of the net displacement = 176 km * cos(70°)
- The y-component (vertical) of the net displacement = 176 km * sin(70°)
- The magnitude of the net displacement = sqrt((176 km * cos(70°))^2 + (176 km * sin(70°))^2)
- The direction of the net displacement = atan(y/x) converted to degrees and adjusted for the quadrant.
Now, you can plug in the values and calculate the answers.