Question

If \[ a=-kv \] represents the variation of acceleration of a particle with velocity, then the time taken to reduce the velocity from \(v\) to \(\dfrac{v}{2}\) is

• A. \[ \frac{\ln 2}{k} \]
• B. \[ \frac{1}{2k} \]
• C. \[ \frac{\ln 2}{2k} \]
• D. \[ \frac{2}{k} \]

Hint:

Whenever acceleration is proportional to velocity, \[ a=-kv, \] the velocity decreases exponentially: \[ v=v_0e^{-kt}. \] The time required to reduce the velocity by half is always \[ t=\frac{\ln 2}{k}. \]

Answer 1

The Correct Option is A

Concept: Acceleration \(a = dv/dt = -kv\). Separate, integrate from \(v\) to \(v/2\): \(\int_v^{v/2} dv/v = -k\int_0^t dt \Rightarrow \ln(1/2) = -kt \Rightarrow t = (\ln 2)/k\).

Step 1: Write the final answer. \(\boxed{t=\frac{\ln 2}{k}}\)