Trouble with a Rocket Propulsion question (Variable Mass & Momentum)

In summary, the conversation discusses a problem involving a rocket and its remaining mass, with the only external force being gravity. The equation for the rocket's velocity is derived and solved using a separable differential equation. However, the initial condition is not satisfied in the result, leading to the conclusion that the lower limit was not evaluated when integrating.
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vparam
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Homework Statement
A fully fueled rocket has a mass of 21,000 kg, of which 15,000 kg is fuel. The burned fuel is spewed out the rear at a rate of 190 kg/s with a speed of 2800 m/s relative to the rocket. If the rocket is fired vertically upward calculate its final velocity at burnout (all fuel used up). Ignore air resistance and assume g is a constant 9.80 m/s^2.
Relevant Equations
M * dv/dt = ∑F_ext + v_rel * dM/dt
I chose to set the upwards direction to be positive and dM/dt = R = 190 kg/s, so I can solve the problem in variable form and plug in. With the only external force being gravity, this gives

M(t) * dv/dt = -M(t) * g + v_rel * R

where M(t) is the remaining mass of the rocket. Rearranging this equation gives:

dv/dt = -((v_rel * R)/M(t)) - g

Since R is constant, M(t) = M_0 - R * t, where M_0 is the initial mass of the rocket. Plugging in gives:

dv/dt = -((v_rel * R)/(M_0 - R * t)) - g.

Solving as a separable differential equation, I arrived at the answer (assuming v = 0 at t = 0):

v(t) = -g * t + v_rel * ln(M_0 - R * t).

However, after plugging in values, I'm not able to get the correct answer. The solution instead has a different equation for v(t):

v(t) = -g * t + v_rel * ln(M/M_0).

Any help about where I could be going wrong with the physical setup or the math of this problem would be much appreciated. Thanks in advance!
 
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  • #2
vparam said:
dv/dt = -((v_rel * R)/(M_0 - R * t)) - g.

Solving as a separable differential equation, I arrived at the answer (assuming v = 0 at t = 0):

v(t) = -g * t + v_rel * ln(M_0 - R * t).
Note that your result for v(t) does not satisfy the initial condition v = 0 at t = 0.

When you integrated ## \dfrac {dt}{M_0 - Rt}## from ##t = 0## to ##t = t##, did you forget to evaluate at the lower limit ##t = 0##?
 
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  • #3
Yes, it looks like that's where I went wrong. Thank you for your help!
 
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1. What is variable mass in rocket propulsion?

Variable mass in rocket propulsion refers to the change in the mass of a rocket as it burns its fuel and propels itself forward. As the fuel is burned and expelled out of the rocket, the mass of the rocket decreases, leading to a change in its momentum and acceleration.

2. How does variable mass affect rocket propulsion?

Variable mass has a significant impact on rocket propulsion as it directly affects the rocket's thrust and acceleration. As the mass decreases, the thrust increases, leading to a higher acceleration. This is known as the rocket's mass ratio, and it is a crucial factor in determining the efficiency and success of a rocket launch.

3. What is momentum in rocket propulsion?

Momentum in rocket propulsion refers to the product of the rocket's mass and its velocity. It is a crucial aspect of rocket propulsion as it determines the force and direction of the rocket's movement. The change in momentum is what allows the rocket to overcome the force of gravity and propel itself into space.

4. How is momentum conserved in rocket propulsion?

In rocket propulsion, momentum is conserved through the principle of action and reaction. As the rocket expels fuel out of its engine, the fuel exerts an equal and opposite force on the rocket, propelling it forward. This ensures that the total momentum of the rocket and its exhaust remains constant, allowing the rocket to continue accelerating.

5. What are the challenges of dealing with variable mass in rocket propulsion?

Dealing with variable mass in rocket propulsion presents several challenges for scientists and engineers. These include accurately predicting the change in mass and its impact on the rocket's performance, designing efficient and reliable propulsion systems, and ensuring the safety of the rocket and its payload during launch and flight.

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