A triathlete swims 1200 m due to east followed by 1500 m due northeast (45o). What is the overall displacement? (magnitude and angle). Calculate the displacement with long-hand calculations.
Long hand? I wonder what that means.
sketch the diagram.
You have two sides, and the included angle. SAS Law of cosines to get the magnitude of the displacement
d^2=1200^2+1500^2+2(1200)(1500)Cos135
solve for displacement magnitude.
Now angle, in my sketch, law of sines:
1500/SinA=d/sin135
you know d, solve for A. But A is measured from East, so the angle from N is 90-A.
thank you bobpursley
To calculate the overall displacement, we first need to break down the two distances into their x and y components based on the given angles.
For the first distance, 1200 m due east, the x component is 1200 m (since it is in the positive x direction), and the y component is 0 m (since there is no movement in the y direction).
For the second distance, 1500 m due northeast at an angle of 45 degrees, we need to determine the x and y components. Since it is at a 45-degree angle, the x and y distances will be equal. To find these distances, we can use trigonometry.
Using the given angle of 45 degrees, we can determine the x and y components using the following trigonometric formulas:
x = distance * cos(angle)
y = distance * sin(angle)
Substituting the values into the formulas, we get:
x = 1500 m * cos(45°) ≈ 1500 m * 0.7071 ≈ 1060.65 m
y = 1500 m * sin(45°) ≈ 1500 m * 0.7071 ≈ 1060.65 m
Now that we have determined the x and y components for both distances, we can add them up to find the net displacement:
Net x displacement = 1200 m + 1060.65 m = 2260.65 m (east)
Net y displacement = 0 m + 1060.65 m = 1060.65 m (north)
To find the magnitude (or overall distance) of the displacement, we can use the Pythagorean theorem:
magnitude = sqrt((x displacement)^2 + (y displacement)^2)
= sqrt((2260.65 m)^2 + (1060.65 m)^2)
≈ sqrt(5119399 m^2)
≈ 2262.59 m
Therefore, the magnitude of the overall displacement is approximately 2262.59 m.
To find the direction or angle of the displacement, we can use trigonometry again:
angle = arctan(y displacement / x displacement)
= arctan(1060.65 m / 2260.65 m)
≈ arctan(0.469)
Using the calculator to find the arctan of 0.469, we get:
angle ≈ 25.52°
Therefore, the angle of the overall displacement is approximately 25.52°.
So, the overall displacement is approximately 2262.59 m at an angle of 25.52° (north of east).