$ ext{The displacement-time graph for the two particles } P_1 ext{ and } P_2 ext{ are straight lines inclined at 30° and 45° with the time axis. The ratio of velocities } V_{P_1} : V_{P_2} ext{ will be :}$

Q: $\text{The displacement-time graph for the two particles } P_1 \text{ and } P_2 \text{ are straight lines inclined at 30° and 45° with the time axis. The ratio of velocities } V_{P_1} : V_{P_2} \text{ will be :}$

$\text{The displacement-time graph for the two particles } P_1 \text{ and } P_2 \text{ are straight lines inclined at 30° and 45° with the time axis. The ratio of velocities } V_{P_1} : V_{P_2} \text{ will be :}$

Question

$	ext{The displacement-time graph for the two particles } P_1 	ext{ and } P_2 	ext{ are straight lines inclined at 30° and 45° with the time axis. The ratio of velocities } V_{P_1} : V_{P_2} 	ext{ will be :}$

Accepted Answer

To determine the ratio of velocities $ V_{P_1} : V_{P_2} $ for the particles $ P_1 $ and $ P_2 $, we need to consider the relationship between the angle of inclination of the displacement-time graph and the velocity of the particles.

1. **Understanding the Relationship:**
   - The displacement-time graph for a particle is a straight line, and the slope of this line represents the velocity of the particle.
   - The slope of a line inclined at an angle $\theta$ with the time axis is given by $\tan(\theta)$.

2. **Calculating Velocities:**
   - For particle $ P_1 $, the line is inclined at $30^\circ$. Therefore, the velocity $ V_{P_1} $ is given by:
     $$
     V_{P_1} = \tan(30^\circ) = \frac{1}{\sqrt{3}}
     $$
   - For particle $ P_2 $, the line is inclined at $45^\circ$. Therefore, the velocity $ V_{P_2} $ is given by:
     $$
     V_{P_2} = \tan(45^\circ) = 1
     $$

3. **Calculating the Ratio:**
   - The ratio of velocities $ V_{P_1} : V_{P_2} $ is:
     $$
     V_{P_1} : V_{P_2} = \frac{1/\sqrt{3}}{1} = \frac{1}{\sqrt{3}} : 1
     $$
   - Simplifying this, we get:
     $$
     V_{P_1} : V_{P_2} = 1 : \sqrt{3}
     $$

4. **Justification for Option 1:**
   - The calculated ratio $ 1 : \sqrt{3} $ matches Option 1, confirming it as the correct answer.

5. **Explanation of Incorrect Options:**
   - **Option 2 ($\sqrt{3} : 1$):** This would imply that $ V_{P_1} $ is greater than $ V_{P_2} $, which contradicts the fact that $\tan(30^\circ) < \tan(45^\circ)$.
   - **Option 3 (1 : 3):** This suggests a much larger difference in velocities than what is calculated, as $\tan(30^\circ)$ is not one-third of $\tan(45^\circ)$.
   - **Option 4 (1 : 1):** This implies equal velocities, which is incorrect because the angles of inclination are different, leading to different slopes and thus different velocities.

In conclusion, the correct ratio of velocities $ V_{P_1} : V_{P_2} $ is indeed $ 1 : \sqrt{3} $, making Option 1 the correct choice.