While exploring a cave, a spelunker starts at the entrance and moves the following distances: 75.0 m north, 250 m east, 210 m at an angle 30.0° north of east, and 150 m south. Find the resultant displacement from the cave entrance.
I know I have to use the formulas Ax = A cos è, Ay = A sin è, A = sq rt((Ax)^2 + (Ay)^2), and è = tan^(-1) (Ay/Ax). But I'm not sure whether I should calculate 125*sin(30) on the calculator because I get a negative number.
è is theta
125sin(30) is 125x0.5 = 62.5
For this sort of problem you need to draw it out on paper and so hard to show on here.
210 m sin(30) + 75 m= 105 m gives the total displacement north. After this he moves south by 150 m so his displacement north is
105m+-150m +75 m = 30 m
[North is the +ve direction)
His displacement east is
250 m + 210 cos 30 = 250 m +182 m = 432 m
so the displacement angle is
tan^-1(30/432) = 3.97 deg (North of east)
and the displacement is
Sqrt(30^2 + 432^2) = 433 m
But check my maths!!
Thanks
To find the resultant displacement from the cave entrance, you will need to calculate the x and y components of each individual displacement and then sum them up.
Let's break down the distances moved:
1. 75.0 m north: This is a purely vertical displacement, so the x component is 0, and the y component is 75.0 m.
2. 250 m east: This is a purely horizontal displacement, so the y component is 0, and the x component is 250 m.
3. 210 m at an angle 30.0° north of east: This displacement has both x and y components. To calculate them, you can use the formulas Ax = A cos(θ) and Ay = A sin(θ).
Ax = 210 m * cos(30°) = 181.499 m
Ay = 210 m * sin(30°) = 105 m
4. 150 m south: This is a purely vertical displacement, so the x component is 0, and the y component is -150 m. (Note that it is negative because it is in the opposite direction of the positive y-axis.)
Now let's sum up the x and y components:
x-component: 0 + 250 m + 181.499 m + 0 = 431.499 m
y-component: 75.0 m + 0 + 105 m - 150 m = 30 m
Now we can use the formula A = √((Ax)^2 + (Ay)^2) to calculate the magnitude of the resultant displacement:
A = √((431.499 m)^2 + (30 m)^2)
Calculating this, we get A ≈ 432.035 m
Finally, we can use the formula θ = tan^(-1)(Ay / Ax) to find the angle or direction of the resultant displacement:
θ = tan^(-1)(30 m / 431.499 m)
Calculating this, we get θ ≈ 4.01°
Therefore, the resultant displacement from the cave entrance is approximately 432.035 m in magnitude and at an angle of approximately 4.01°.