Question

A particle is moving along the X-axis. Its acceleration at time \( t \) is proportional to its velocity at that time. Find the differential equation of the motion of the particle.

Hint:

When acceleration is proportional to velocity, the differential equation is: \[ \frac{dv}{dt} = k v. \]

Answer 1

Step 1: Relating Acceleration to Velocity
Given that acceleration \( a \) is proportional to velocity \( v \), we have: \[ a = k v \] Since acceleration is the rate of change of velocity, we can write: \[ \frac{dv}{dt} = k v. \] Step 2: Formulating the Differential Equation
Rearrange the equation to separate the variables: \[ \frac{dv}{v} = k \, dt. \] This is the desired first-order differential equation.