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

Step 1: Understanding the Concept:
For a displacement versus time (\(s-t\)) graph, the velocity of the particle is defined as the rate of change of displacement with respect to time.
Graphically, this is represented by the slope of the line.
Key Formula or Approach:
The slope of a straight line on a Cartesian plane is given by \(\tan \theta\), where \(\theta\) is the angle the line makes with the positive x-axis.
Velocity (\(v\)) = \(\frac{ds}{dt}\) = Slope = \(\tan \theta\).
Step 2: Detailed Explanation:
Let \(v_1\) be the velocity of the first particle and \(v_2\) be the velocity of the second particle.
1. The first particle makes an angle of \(30^\circ\) with the time axis.
\[ v_1 = \tan 30^\circ = \frac{1}{\sqrt{3}} \]
2. The second particle makes an angle of \(45^\circ\) with the time axis.
\[ v_2 = \tan 45^\circ = 1 \]
3. To find the ratio \(v_1 : v_2\):
\[ \frac{v_1}{v_2} = \frac{1/\sqrt{3}}{1} = \frac{1}{\sqrt{3}} \]
The ratio is therefore \(1 : \sqrt{3}\).
Step 3: Final Answer:
The ratio of the respective velocities is \(1 : \sqrt{3}\).