Question

Two points move in the same straight line starting at the same moment from the same point. One moves with velocity \(u\) and the other with acceleration \(f\). The greatest distance between them is:

• A. \( \frac{u}{f} \)
• B. \( \frac{u^2}{2f} \)
• C. \( \frac{f}{2u^2} \)
• D. \( \frac{f}{u^2} \)

Hint:

Maximum separation occurs when relative velocity becomes zero.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The distance between two particles is maximum when their relative velocity is zero, i.e., when the instantaneous velocity of the accelerating particle equals the constant velocity of the first particle.
Step 2: Key Formula or Approach:
Displacement of particle 1: \(x_{1} = ut\)
Displacement of particle 2: \(x_{2} = \frac{1}{2}ft^{2}\)
Velocity of particle 2: \(v = ft\)
: Detailed Explanation:
Equating velocities to find the time of maximum separation:
\[ ft = u \implies t = \frac{u}{f} \]
At this time, calculate the positions:
\[ x_{1} = u \left( \frac{u}{f} \right) = \frac{u^{2}}{f} \]
\[ x_{2} = \frac{1}{2}f \left( \frac{u}{f} \right)^{2} = \frac{u^{2}}{2f} \]
The greatest distance \(D\) is:
\[ D = x_{1} - x_{2} = \frac{u^{2}}{f} - \frac{u^{2}}{2f} = \frac{u^{2}}{2f} \]
Step 3: Final Answer:
The maximum separation is \(\frac{u^{2}}{2f}\).