Answer to Question #237784 in Mechanics | Relativity for Amy

Question

Let A= (3.0 m, 200 south of east), B = (2.0 m, north), and C= (5.0 m, 700 south of west).

a. Draw and label A , B, and C with their tails at the origin. Use a coordinate system with the x-axis to the east.

b. b. Write A, B, and C in component form, using unit vectors.

c. c. Find the magnitude and the direction of D = A + B+ C.

Answer 1

Solution.

x = R*cos(θ)

y = R*sin(θ)

a) Let's define east as the positive x-axis, north as the positive y-axis.

Here we have:

A = (3.0m, 20° sout of east) = (3.0m, -20°)

B = (2.0m, north) = (2.0m, 90°)

C = (5.0m, 70° sout of west) = (5.0m, 250°)

The graph of the 3 vectors can be seen in the image below.

b) We want to write A, B, and C in component form.

We start with A = (3.0m, -20°)

x = 3.0m*cos(-20°) = 2.82m;

y = 3.0m*sin(-20°) = -1.03m;

A = (2.82m, -1.03m);

B = (2.0m, 90°);

x = 2.0m*cos(90°) = 0m;

y = 2.0m*sin(90°) = 2.0m;

B = (0m, 2.0m);

C = (5.0m, 250°);

x = 5.0m*cos(250°) = -1.7m;

y = 5.0m*sin(250°) = -4.7m;

C = (-4.7m, -1.7m);

c) Now we want to find the magnitude and direction of:

D = A + B + C

 = (2.82m, -1.03m) + (0m, 2.0m) + (-4.7m, -1.7m)

 = (2.82m + 0m - 4.7m, -1.03m + 2.0m - 1.7m)

 = (-1.88m, -0.73m)

The magnitude is given by:

$D=\sqrt{(-1.88)^2+(-0.73)^2}=2,02m;$

To find the direction, or the angle, we can write:

θ = Atan(y/x) = Atan(-0.73m/-1.88m) = 21.22°

Then we can write D as:

D = (2.02m, 21.22°).

Answer: D = (2.02m, 21.22°).