Problem 9 - Master Resource Book in JEE Main Physics

Question

One of the rectangular components of a velocity of \(60 \mathrm{kmh}^{-1}\) is \(30 \mathrm{kmh}^{-1}\). The other rectangular component is (a) \(30 \mathrm{kmh}^{-1}\) (b) \(30 \sqrt{3} \mathrm{kmh}^{-1}\) (c) \(30 \sqrt{2} \mathrm{kmh}^{-1}\) (d) zero

Step-by-Step Solution

Answer

The other rectangular component is \(30\sqrt{3} \mathrm{kmh}^{-1}\) (option b).

Step 1: Understand the Problem

We have a vector with a magnitude of 60 km/h. One of its rectangular components is given as 30 km/h. We need to find the magnitude of the other component and match it with one of the given options.

Step 2: Use the Pythagorean Theorem

For a vector with one component being 30 km/h and the vector's magnitude as 60 km/h, use the Pythagorean theorem: \( c^2 = a^2 + b^2 \), where \( c \) is the magnitude of the vector, and \( a \) and \( b \) are the components.

Step 3: Substitute Known Values

Substitute the known values into the equation: \( 60^2 = 30^2 + b^2 \). This simplifies to \( 3600 = 900 + b^2 \).

Step 4: Solve for the Unknown Component

Rearrange the formula to solve for \( b \): \( b^2 = 3600 - 900 = 2700 \). Therefore \( b = \sqrt{2700} \).

Step 5: Simplify the Solution

Simplify \( b = \sqrt{2700} \). We can express 2700 as \( 900 \times 3 \) (since 900 is a perfect square, \( 900 = 30^2 \)), so \( b = \sqrt{30^2 \times 3} = 30\sqrt{3} \).

Step 6: Confirm with Options

Now evaluate the options: (a) 30 km/h, (b) \( 30 \sqrt{3} \) km/h, (c) \( 30 \sqrt{2} \) km/h, (d) zero. The component \( b = 30\sqrt{3} \) km/h matches option (b).

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry. It's essential for understanding relationships in right triangles. It states that the square of the hypotenuse (the longest side of a right triangle) is equal to the sum of the squares of the other two sides. The mathematical form of this is:

• \( c^2 = a^2 + b^2 \)

Here, \( c \) represents the hypotenuse, while \( a \) and \( b \) are the triangle's legs. This theorem allows us to find missing side lengths, as long as we know two sides. Apply this to vectors, and each vector can be seen as a hypotenuse of a right triangle with its components forming the legs.
When a magnitude (like our vector's 60 km/h) and one component (30 km/h) are known, plug these into the equation to solve for the other component.
The theorem provides a reliable way to handle vector decomposition via simple math.

velocity vectors

Velocity vectors describe speed and direction. They're vital in physics and engineering and help depict how fast an object moves in a direction.
Vectors have both magnitude (speed) and direction. Imagine it as an arrow: the arrow’s length indicates speed, while its direction shows the movement path.
In the exercise, we explore a velocity vector of 60 km/h. This vector can be split into two perpendicular components, like an arrow stretched into two smaller arrows. These components provide an easier way to analyze motion, especially in physics problems that involve different directions.

• Magnitude: This tells you the size or length of the vector (like the 60 km/h).
• Components: These are the projections on the x and y axes. Think of them as forces pulling in each axis' direction.

By breaking down velocity into components, understanding motion, especially in two dimensions, becomes straightforward.

rectangular components

Rectangular components are essential in vector analysis. They allow a vector's magnitude and direction to be broken down into perpendicular parts, often along the x and y axes.
When you work with vectors, often it's easier to handle calculations using these components because they turn the problem into simple right triangle math.
For example, the given velocity vector (60 km/h) splits into a 30 km/h component and another that we need to find using our formulas.
This is practical as rectangular components:

• Make it easier to add, subtract, and manipulate vectors mathematically.
• Help visualize the vector's influence in two directions.
• Offer a way to simply use the Pythagorean theorem to resolve complex vector problems.

Essentially, by finding rectangular components like \( 30 \sqrt{3} \) km/h, problems that initially seem complex become manageable. This method is widely used in physics and engineering for resolving forces and velocities.