Question

your dog is running around the grass in your back yard. he undergoes successive displacements 3.62 m south, 8.10 m northeast, and 14.6 m west. what is the resultant displacement? (take the x direction to be east.)

Answer 1

The displacement can be represented as vectors in the x and y direction.

The displacement 3.62 m south can be represented as a vector of magnitude 3.62 m and direction -90°.

The displacement 8.10 m northeast can be represented as a vector of magnitude 8.10 m and direction 45°.

The displacement 14.6 m west can be represented as a vector of magnitude 14.6 m and direction -180°.

We can find the x and y components of each displacement vector, and then add those components to find the x and y components of the resultant displacement.

x component of the first displacement: 3.62 * cos(-90°) = 0 m

y component of the first displacement: 3.62 * sin(-90°) = -3.62 m

x component of the second displacement: 8.10 * cos(45°) = 5.65 m

y component of the second displacement: 8.10 * sin(45°) = 5.65 m

x component of the third displacement: 14.6 * cos(-180°) = -14.6 m

y component of the third displacement: 14.6 * sin(-180°) = 0 m

The x and y components of the resultant displacement are the sum of the x and y components of each individual displacement:

x component of the resultant displacement: 0 + 5.65 - 14.6 = -8.95 m

y component of the resultant displacement: -3.62 + 5.65 + 0 = 2.03 m

We can use the Pythagorean theorem to find the magnitude and direction of the resultant displacement:

resultant displacement = √(x^2 + y^2) = √(-8.95^2 + 2.03^2) = 9.06 m

direction = tan^-1(y/x) = tan^-1(2.03/-8.95) = 125.3°

So, the resultant displacement is 9.06 m at an angle of 125.3° with respect to the x-axis (east).

Step-by-step explanation: