Question

In a rocket-propulsion problem the mass is variable. Another such problem is a raindrop falling through a cloud of small water droplets. Some of these small droplets adhere to the raindrop, thereby increasing its mass as it falls. The force on the raindrop is

Fext=dpdt=mdvdt+vdmdt
Suppose the mass of the raindrop depends on the distance x that it has fallen. Then m=kx, wherek is a constant, and dm/dt=kv. This gives, since Fext=mg,
mg=mdvdt+v(kv)
Or, dividing by k,
xg=xdvdt+v2
This is a differential equation that has a solution of the form v=at, where a is the acceleration and is constant. Take the initial velocity of the raindrop to be zero.
A) Using the proposed solution for v, find the acceleration a.
B) Find the distance the raindrop has fallen in t = 2.00s.
C) Given that k = 2.00g/m, find the mass of the raindrop at t = 2.00s.

Answer 1

a) a = g / 3

b) x (3.0) = 14.7 m

c) m (3.0) = 29.4 g

Step-by-step explanation:

Given:-

- The following differential equation for (x) the distance a rain drop has fallen has the form:

- Where, v = Speed of the raindrop

- Proposed solution to given ODE:

v = a*t

Where, a = acceleration of raindrop

Find:-

(a) Using the proposed solution for v find the acceleration a.

(b) Find the distance the raindrop has fallen in t = 3.00 s.

(c) Given that k = 2.00 g/m, find the mass of the raindrop at t = 3.00 s.

Solution:-

- We know that acceleration (a) is the first derivative of velocity (v):

a = dv / dt ... Eq 1

- Similarly, we know that velocity (v) is the first derivative of displacement (x):

v = dx / dt , v = a*t ... proposed solution (Eq 2)

v .dt = dx = a*t . dt

- integrate both sides:

∫a*t . dt = ∫dt

x = 0.5*a*t^2 ... Eq 3

- Substitute Eq1 , 2 , 3 into the given ODE:

0.5*a*t^2*g = 0.5*a^2 t^2 + a^2 t^2

= 1.5 a^2 t^2

a = g / 3

- Using the acceleration of raindrop (a) and t = 3.00 second and plug into Eq 3:

x (t) = 0.5*a*t^2

x (t = 3.0) = 0.5*9.81*3^2 / 3

x (3.0) = 14.7 m

- Using the relation of mass given, and k = 2.00 g/m, determine the mass of raindrop at time t = 3.0 s:

m (t) = k*x (t)

m (3.0) = 2.00*x(3.0)

m (3.0) = 2.00*14.7

m (3.0) = 29.4 g