find mass of a rocket as a function of time,if it moves with constant acceleration a in absence of external forces.the gas escapes with a constant velocity u relative to the rocket and its mass initialy was m'
The rate of mass release, and the thrust F, would have to vary continuously to maintain a constant acceleration. That is not the way rockets usually work, but it is possible.
a = F(t)/m(t)= (u *dm/dt)/m
Integrate and solve the differentuial equation with m = m' at t = 0
u ln (m'/m) = a*t
(a/u)t = ln(m'/m)
e^(at/u) = m'/m
m = m'*e^(-at/u)
To find the mass of the rocket as a function of time, you need to consider the principle of conservation of momentum. When the gas escapes from the rocket, it exerts an opposite force on the rocket according to Newton's third law of motion. This force causes the rocket to undergo a constant acceleration.
Let's denote the mass of the rocket at any given time, t, as m(t). According to the principle of conservation of momentum, we can write:
m(t) * v(t) = (m' - dm(t)) * (v(t) + u)
Where:
- m(t) is the mass of the rocket at time t.
- v(t) is the velocity of the rocket at time t.
- m' is the initial mass of the rocket.
- dm(t) is the infinitesimal mass of the rocket ejected during the infinitesimal time interval dt.
- u is the constant velocity at which gas escapes relative to the rocket.
As the gas escapes, the mass of the rocket decreases, causing dm(t) to be negative. Integrating the above equation gives:
∫[m(t)/ (m' - m(t))] * dm(t) = ∫[v(t) + u] dt
To solve this equation, we need to determine the relationship between the rocket's velocity and its mass. This involves knowing the specific acceleration function, a(t). The acceleration is related to the change in velocity with time:
a(t) = dv(t) / dt
If the acceleration is constant, then a(t) = a.
Now we can integrate both sides of the equation to find the mass as a function of time:
∫[m(t) / (m' - m(t))] * dm(t) = ∫[v(t) + u] dt
This equation can be solved by using an appropriate integration technique, depending on the specific values of a(t) and u.
Please note that the specific solution to this problem will depend on the specific values and functions involved.