Question

1. a kayak is traveling across a pond at a speed of 8 meters per second in the direction of s 67° w. give the speed of the kayak in component form.

Explanation:

Step1: Resolve the velocity vector into components

We know that if the magnitude of the velocity is $v = 8$ m/s and the angle $\theta=67^{\circ}$ west of south. The south - component of the velocity $v_y$ and west - component of the velocity $v_x$ can be found using trigonometry. The south - component of the velocity is given by $v_y=v\cos\theta$ and the west - component is given by $v_x = v\sin\theta$.

Step2: Calculate the south - component

$v_y=8\cos(67^{\circ})$. Since $\cos(67^{\circ})\approx0.391$, then $v_y = 8\times0.391 = 3.128$ m/s south.

Step3: Calculate the west - component

$v_x=8\sin(67^{\circ})$. Since $\sin(67^{\circ})\approx0.921$, then $v_x = 8\times0.921=7.368$ m/s west.
In component form, the velocity vector $\vec{v}=( - 7.368\hat{i}-3.128\hat{j})$ m/s, where the negative $x$ - direction is west and negative $y$ - direction is south.

Answer:

The velocity of the kayak in component form is approximately $(-7.37\hat{i}-3.13\hat{j})$ m/s (rounded to two decimal places)