Explanation

The relativistic length contraction formula is:

L = L₀√(1 - v²/c²)

Given L = L₀/2:

L₀/2 = L₀√(1 - v²/c²)

1/2 = √(1 - v²/c²)

1/4 = 1 - v²/c²

v²/c² = 3/4

v = √(3/4) * c = √3/2 * c

The answer is v = √3/2c.

Explanation

Non-terminating repeating decimals can be expressed as fractions, so they are rational numbers.

In contrast, irrational numbers have non-terminating, non-repeating decimal expansions.

Explanation

Fluids are substances that can flow — this includes both liquids and gases.

Unlike solids, fluids have no fixed shape and take the shape of their container.

Explanation

Gamma rays consist of radiations similar to x-rays.

They are high-energy electromagnetic waves with no mass or charge.

Explanation

Given dt/dx = xy^(1/2)

Separate variables:

dt/y^(1/2) = x dx

Integrate both sides:

∫y^(-1/2) dy = ∫x dx

2y^(1/2) = (x²)/2 + C

y^(1/2) = x²/4 + C'

Square both sides:

y = (x²/4 + C')²

For simplicity, let's assume C' = 0:

y = (x²/4)² = x⁴/16

Explanation

حدیث میں نبی ﷺ نے فرمایا کہ بدن (جسم) کی زکوٰۃ روزہ ہے، یعنی روزہ بدن کو پاکیزگی اور روحانی طہارت دیتا ہے۔

روزہ جسم کو عبادت اور ضبطِ نفس کا ذریعہ بناتا ہے، اس لیے اسے بدن کی زکوٰۃ قرار دیا گیا ہے۔

Explanation

نماز میں آخری قعدہ تشہد (التحیات) مکمل پڑھنے تک واجب ہے۔

اگر تشہد مکمل پڑھے بغیر اٹھا جائے تو سجدۂ سہو واجب ہوتا ہے۔

Explanation

At 45°, the sine and cosine of the angle are equal (sin 45° = cos 45° ≈ 0.707), so the x and y components of the force are equal.

This is because Fx = Fcosθ and Fy = Fsinθ; they are equal only when cosθ = sinθ.