What are the direction and magnitude of your total displacement if you have traveled due west with a speed of 25.0 m/s for 125 s, then due south at 19.0 m/s for 66.0 s?
Picture a right angled triangle with one side AB being 25 x 125 metres long to the west; the other side BC being 19 x 66 metres long to the south. The displacement is the length of the hypotenuse AC; the direction being between 180 degrees and west. Sketch it out, then do the trig.
To determine the total displacement, we need to find the vector sum of the individual displacements in both directions.
Let's start by calculating the westward displacement.
Displacement (west) = velocity (west) × time (west)
= 25.0 m/s × 125 s
= 3125 m west
Next, let's calculate the southward displacement.
Displacement (south) = velocity (south) × time (south)
= 19.0 m/s × 66.0 s
= 1254 m south
Now, to find the total displacement, we need to find the vector sum of these two displacements. Since they are in perpendicular directions (west and south), we can calculate their magnitude and direction using the Pythagorean theorem and trigonometry.
Magnitude of the total displacement:
Magnitude = √((Displacement (west))^2 + (Displacement (south))^2)
= √((3125 m)^2 + (1254 m)^2)
≈ 3364 m
Direction of the total displacement:
To find the direction, we can use the inverse tangent function (tan^(-1)) to calculate the angle between the total displacement and the west direction.
Angle = tan^(-1)((Displacement (south))/(Displacement (west)))
= tan^(-1)((1254 m)/(3125 m))
≈ 21.1° south of west
Therefore, the total displacement is approximately 3364 m at an angle of 21.1° south of west.