One of the rectangular components of a velocity of `60" km" h ^(-1) " is "30 " km " h ^(-1)` . Find other rectangular component ?

Options

A. `15sqrt(3)km h ^(-1)`
B. `30sqrt(6)km h ^(-1)`
C. `30sqrt(3)km h ^(-1)`
D. `15sqrt(6)km h ^(-1)`

Answer

To find the other rectangular component of the velocity given that one component is \(30 \, \text{km/h}\) and the total velocity is \(60 \, \text{km/h}\), we can follow these steps:

### Step 1: Understand the Components
We have a velocity vector \( \mathbf{v} \) that can be broken down into its rectangular components along the x-axis and y-axis. Let's denote:
- \( v_x \) as the x-component (unknown)
- \( v_y = 30 \, \text{km/h} \) as the y-component (given)

### Step 2: Use the Pythagorean Theorem
The magnitude of the velocity vector can be expressed using the Pythagorean theorem:
\[ |\mathbf{v}| = \sqrt{v_x^2 + v_y^2} \]
Given that the magnitude of the velocity \( |\mathbf{v}| = 60 \, \text{km/h} \), we can write:
\[ 60 = \sqrt{v_x^2 + 30^2} \]

### Step 3: Square Both Sides
To eliminate the square root, we square both sides of the equation:
\[ 60^2 = v_x^2 + 30^2 \]
This simplifies to:
\[ 3600 = v_x^2 + 900 \]

### Step 4: Isolate \( v_x^2 \)
Now, we will isolate \( v_x^2 \) by subtracting \(900\) from both sides:
\[ v_x^2 = 3600 - 900 \]
\[ v_x^2 = 2700 \]

### Step 5: Solve for \( v_x \)
Taking the square root of both sides gives:
\[ v_x = \sqrt{2700} \]
We can simplify \( \sqrt{2700} \):
\[ \sqrt{2700} = \sqrt{900 \times 3} = \sqrt{900} \times \sqrt{3} = 30\sqrt{3} \]

### Conclusion
Thus, the other rectangular component \( v_x \) is:
\[ v_x = 30\sqrt{3} \, \text{km/h} \]

### Final Answer
The other rectangular component of the velocity is \( 30\sqrt{3} \, \text{km/h} \).

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