Explore BrainMass

Explore BrainMass

    Motion with variable mass

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    A spherical raindrop of radius a, falls from rest under gravity. It falls through a stationary cloud so that, because of condensation, its radius increases with time at a constant rate k. Find the distance fallen by the raindrop after time t.

    © BrainMass Inc. brainmass.com December 24, 2021, 11:38 pm ad1c9bdddf
    https://brainmass.com/physics/newtons-second-law/motion-variable-mass-587927

    SOLUTION This solution is FREE courtesy of BrainMass!

    A spherical raindrop of radius a, falls from rest under gravity. It falls through a stationary cloud so that, because of condensation, its radius increases with time at a constant rate k. Find the distance fallen by the raindrop after time t

    Let the radius of the drop at time t be r

    The rate of increase of the radius is given by

    As the initial radius is a and it increases in time t by Kt the radius r at time t will be

    r = a + K*t
    The volume of the drop at time t will be
    And hence its mass will be m =
    Here is the density of water.

    Now according to the second law of motion the rate change of momentum is equal to the force applied and the only force acting on the drop is the gravity (neglecting air resistance).

    As the mass of the drop is variable, hence taking downward direction positive, we can write

    Where v is the velocity of the drop at time t (initial velocity is zero)

    Substituting the value of mass from above we get

    Or

    Integrating this equation wrt t we get

    Here C is the constant of integration.
    As at t = 0; v = 0 substituting in the equation we have

    Substituting this in above equation we have

    Or

    Or ------------------ (2)
    Now if the distance fallen in time t is x then the velocity is given by v = dx/dt and hence

    Integrating wrt to t we have

    Where C' is constant of integration.
    Or

    Now as initially at t = 0; x = 0 gives

    Substituting in equation above we get

    Or
    Or
    Or
    Or

    This is the required distance.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 11:38 pm ad1c9bdddf>
    https://brainmass.com/physics/newtons-second-law/motion-variable-mass-587927

    Attachments

    ADVERTISEMENT