Help with rocket's final velocity problem

[MTitus]

The Background:

This is a question that is worth extra credit for my Honors 131 Physics class in college. This course is the first calculus-based physics course that I have taken so I've been having trouble with integrating calculus into some physics equations.

Thus far we have learned about interactions, vectors, momentum and impulse, particles and systems, and momentum conservation laws and principles.

The Question:

Consider a rocket in deep space whose empty mass is 5200kg that can carry 52,000kg of propellant. If the rocket engine can eject 1300kg/s of propellant from its nozzle at a speed of 3300m/s relative to the rocket, and if the rocket starts at rest, what is its final speed? This problem is difficult because the ship's mass changes constantly as the proppellant is ejected. Either solve the problem mathematically using calculus, taking the constantly decreasing mass of the rocket into account, or numerically using a calculator or a computer program. If you do the former, note that conservation of momentum implies that hte rate of change of the total momentum of the rocket and the exhaust must be zero. If you do the latter, pretend that each 1s the rocket ejects a 1300-kg "chunk" of propellant, and use conservation of momentum to compute the final speed of the rocket (including the remaining propellant). You will have to repeat this calculation for each second that the engines fire.

** My professor also added on our homework sheet: This is a more difficult (but very interesting) problem to challenge the most advanced students! It requires solving a simple differential equation in time. To obtain this differential equation, take into account that the burning fuel not only delivers momentum to the payload, but carries itself momentum which is "lost" to the payload when the burnt fuel leaves the rocket.

** After talking to my professor, I know that I must solve it mathematically using calculus in order to receive credit.

My Problem History:

I know I need to find velocity so I have the equation
v(t) = p(t)/m(t).

I have found that m(t) is equal to
m(t) = (52000kg - 1300kg/s * t) + 5200kg.

Finding p(t) has proved to be harder for me than finding m(t). My first equation that I found for p(t) was
p(t) = 1300kg/s * 3300m/s * t.

Simplifying and plugging in numbers (using 40 seconds as the time it takes for the entire proppellant to be expelled) I found that
v(40) = [4290000kg-m/s * (40s)]/[57200kg - 1300kg/s * (40s)]
v(40) = 33000m/s

This will not work however because momentum is taken away from the overall momentum of the rocket when the mass of proppellant is expelled from the rocket, and this, is where I am stuck. I've thought about doing something like
p(t) = [1300kg/s * 3300m/s * t] - 1300kg/s * t.

That won't work either... I'm really stuck with this and if someone could point me in the right direction for finding how the momentum equation looks like, that would be amazing and greatly appreciated. Thanks for any help in advance!

 

[Hootenanny]

This page may be useful; http://ed-thelen.org/rocket-eq.html

 

[kyle8921]

Dear student,

Thank you for reaching out for assistance with your rocket's final velocity problem. It is great to see that you are taking on this challenging extra credit problem and seeking help when needed.

First, let's start by breaking down the problem and understanding what we know and what we need to find. We know the mass of the rocket (5200kg) and the mass of the propellant it can carry (52,000kg). We also know the rate at which the propellant is being ejected (1300kg/s) and the speed at which it is being ejected (3300m/s). Finally, we know that the rocket starts at rest and we need to find its final velocity.

To solve this problem, we need to use the conservation of momentum principle. This principle states that the total momentum of a system remains constant unless acted upon by an external force. In this case, the system is the rocket and the propellant, and the external force is the ejection of the propellant.

Let's start by defining our variables:
- m(t) = mass of the rocket at time t
- v(t) = velocity of the rocket at time t
- p(t) = momentum of the rocket at time t

We can also define the momentum of the ejected propellant as p_p(t). Since the propellant is being ejected at a constant rate, we can express its momentum as:
p_p(t) = (1300kg/s * 3300m/s) * t = 4290000kg-m/s * t

Now, using the conservation of momentum principle, we can set up the following equation:
p(t) + p_p(t) = constant

Since we know that the initial momentum of the rocket is zero (since it starts at rest), we can simplify the equation to:
p(t) = -p_p(t)

Substituting in our expression for p_p(t), we get:
p(t) = -4290000kg-m/s * t

Now, we can use the equation for momentum (p = mv) to find the velocity of the rocket at any given time t:
v(t) = p(t)/m(t)

Substituting in our expression for p(t) and m(t) from the problem, we get:
v(t) = (-4290000kg-m/s * t)/(52000kg - 1300kg/s * t + 5200kg)

Since we are