Question

The displacement time graphs of two moving particles make angles of \(30\degree\) and \(45\degree\) with the x-axis as shown in the figure. The ratio of their respective velocity is

• A. \(\sqrt3:1\)
• B. \(1:1\)
• C. \(1:2\)
• D. \(1:\sqrt3\)

Answer 1

The Correct Option is D

The displacement-time graph of a moving particle indicates its velocity by the slope of the line. The formula for velocity \(v\) is given by the slope \(\tan(\theta)\), where \(\theta\) is the angle made with the x-axis. Thus, the velocities of two particles can be determined by the angles provided.

For the particle with a \(30\degree\) angle, the velocity is:

\(v_1 = \tan(30\degree) = \frac{\sqrt{3}}{3}\)

For the particle with a \(45\degree\) angle, the velocity is:

\(v_2 = \tan(45\degree) = 1\)

The ratio of their respective velocities is:

\(\text{Ratio} = \frac{v_1}{v_2} = \frac{\frac{\sqrt{3}}{3}}{1} = \frac{\sqrt{3}}{3}\)

To express this as a ratio of whole numbers, we simplify:

\(\frac{\sqrt{3}}{3} : 1 = 1 : \sqrt{3}\)

Thus, the correct ratio of their respective velocities is \(1:\sqrt3\).

Answer 2

In a displacement-time graph, the slope of the line represents the velocity of the object.

The slope (velocity) is given by:

v = tan(θ), where θ is the angle the line makes with the time (x) axis.

For the two particles:

• θ₁ = 30°, so v₁ = tan(30°) = \(\frac {1}{\sqrt 3}\)
• θ₂ = 45°, so v₂ = tan(45°) = 1

Ratio of their velocities:
v₁ : v₂ =\((\frac {1}{\sqrt 3})\) : 1 = \(1 : \sqrt3\)

The correct option is (D): \(1:\sqrt3\)