Solving the Tsiolokovsky Eqn for Velocity w/Time

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In summary, the conversation discusses the derivation of the Tsiolokovsky equation to find velocity with respect to time. The equation takes into account the rocket's loss of mass and involves replacing the force and mass terms. The integral of acceleration is used to find velocity, but there is a question about the initial velocity when t = 0. It is suggested that the constant of integration is needed to fix this issue.
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Hello! I'm doing a little derivation of the Tsiolokovsky equation where I'm trying to find velocity with respect to time, here's what I got so far:

F=ma, a = F/m

Here I replace the force term and the mass term, taking into account that the rocket is losing mass:

##a = \frac{v_e\cdot \dot{x}}{m_0 - \dot{x}\cdot t}##

where:
v = exhaust velocity
m dot = mass flow rate
m naught = initial mass

After this I take the integral of acceleration to get velocity, it's a pretty easy one since the the top two terms, mass flow rate and exhaust velocity, are both constants:

##\int \frac{v_e\cdot\dot{x}}{m_0 - \dot{x}\cdot t} = -v_e\cdot ln(m_0- \dot{x}\cdot t) + c##

Which is the velocity. My problem with this though is that when t = 0, velocity is ##-v_e \cdot ln(m_0)## which doesn't make any sense, right from the start there is an instantaneous velocity? Maybe the constant of integration is suppose to fix that? Any help would be appreciated.
 
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  • #2
if i am correct then , yes , the constant of integration is there to fix that. i haven't checked your calculations as they look fine. c is supposed to be (ve * ln(mo ). if you evaluate the definite integral of the function between t = 0 and t = ti (for any ti) then you'll see that the velocity function
v = -ve * ln(mo - m'ti) + ve*ln(mo)
 
  • #3
Ah OK, I was just beginning to suspect that I as I was writing that post. Thanks for the help.
 

What is the Tsiolkovsky Equation?

The Tsiolkovsky Equation, also known as the Rocket Equation, is a mathematical formula that relates the change in velocity of a rocket to the amount of propellant used and the efficiency of the rocket engine.

Why is it important to solve the Tsiolkovsky Equation for velocity with time?

Solving the Tsiolkovsky Equation for velocity with time allows scientists and engineers to accurately calculate the velocity of a rocket at any given time during its flight. This information is crucial for planning and optimizing space missions.

What are the variables in the Tsiolkovsky Equation?

The variables in the Tsiolkovsky Equation are the initial velocity of the rocket (v0), the final velocity of the rocket (vf), the mass of the rocket (m), the mass of the propellant (mp), and the effective exhaust velocity (Ve).

How do you solve the Tsiolkovsky Equation for velocity with time?

To solve the Tsiolkovsky Equation for velocity with time, you can use the following formula: vf = v0 + Ve * ln(m0/m), where m0 is the initial total mass of the rocket (m + mp) and m is the final total mass of the rocket (m + mp - mt), where mt is the mass of the propellant consumed at a particular time t.

What are the limitations of the Tsiolkovsky Equation?

The Tsiolkovsky Equation assumes that the rocket is traveling in a vacuum and does not account for external forces such as air resistance. It also assumes that the exhaust velocity remains constant throughout the entire flight, which may not be the case in reality.

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