Questions: As shown in the diagram below, a soccer player kicks a ball directly toward the goal along a displacement vector r, which has a length of 8.2 m and a direction 30 degrees east of south. Alternatively, she could have kicked it to a second player standing directly in front of the goal along a displacement ry' who could have then kicked it to the goal along a displacement vector rx' as shown. Find the magnitudes and directions of rx and ry. (a) rx magnitude m direction -- Select-- (b) ry magnitude m direction ---Select---

As shown in the diagram below, a soccer player kicks a ball directly toward the goal along a displacement vector r, which has a length of 8.2 m and a direction 30 degrees east of south. Alternatively, she could have kicked it to a second player standing directly in front of the goal along a displacement ry' who could have then kicked it to the goal along a displacement vector rx' as shown. Find the magnitudes and directions of rx and ry. (a) rx magnitude m direction -- Select-- (b) ry magnitude m direction ---Select---
Transcript text: As shown in the diagram below, a soccer player kicks a ball directly toward the goal along a displacement vector $\mathbf{r}$, which has a length of 8.2 m and a direction 30 degrees east of south. Alternatively, she could have kicked it to a second player standing directly in front of the goal along a displacement $r_{y^{\prime}}$ who could have then kicked it to the goal along a displacement vector $r_{x^{\prime}}$ as shown. Find the magnitudes and directions of $r_{x}$ and $r_{y}$. (a) $r_{x}$ magnitude $\square$ m direction -- Select-- (b) $r_{y}$ magnitude $\square$ m direction ---Select---
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Solution

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Solution Steps

Step 1: Identify the given information
  • The displacement vector \( \mathbf{r} \) has a length of 8.2 m and a direction of 30 degrees east of south.
  • We need to find the magnitudes and directions of \( \mathbf{r_x} \) and \( \mathbf{r_y} \).
Step 2: Resolve the vector \( \mathbf{r} \) into its components
  • The vector \( \mathbf{r} \) can be resolved into its x-component (\( \mathbf{r_x} \)) and y-component (\( \mathbf{r_y} \)).
  • Using trigonometry:
    • \( r_x = r \sin(\theta) \)
    • \( r_y = r \cos(\theta) \)
    • Where \( r = 8.2 \) m and \( \theta = 30^\circ \).
Step 3: Calculate the x-component (\( \mathbf{r_x} \))
  • \( r_x = 8.2 \sin(30^\circ) \)
  • \( \sin(30^\circ) = 0.5 \)
  • \( r_x = 8.2 \times 0.5 = 4.1 \) m
Step 4: Calculate the y-component (\( \mathbf{r_y} \))
  • \( r_y = 8.2 \cos(30^\circ) \)
  • \( \cos(30^\circ) = \sqrt{3}/2 \approx 0.866 \)
  • \( r_y = 8.2 \times 0.866 \approx 7.1 \) m
Step 5: Determine the directions of \( \mathbf{r_x} \) and \( \mathbf{r_y} \)
  • \( \mathbf{r_x} \) is directed east.
  • \( \mathbf{r_y} \) is directed south.

Final Answer

  • (a) \( \mathbf{r_x} \)
    • Magnitude: 4.1 m
    • Direction: East
  • (b) \( \mathbf{r_y} \)
    • Magnitude: 7.1 m
    • Direction: South
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