i need help figuring out what formaula to use for finding the final velocity of a rocket.

Model rocket engines are sized by thrust, thrust duration, and total impulse, among other characteristics. A size C5 model rocket engine has an average thrust of 5.26 N, a fuel mass of 12.7 g, and an initial mass of 25.5 g. The duration of its burn is 1.90 s.
If this engine is placed in a rocket body of mass 57.5 g, and if the rocket is fired in outer space, what is the final velocity of the rocket? Assume the fuel burns at a constant rate.

You need to derive or find the "rocket" formula for the velocity of an object that has a fixed force T applied while its mass is decreasing. Calculus is needed to derive it.

You can find it at

http://www.ebtx.com/mars/rocketeq.htm

You will need the rocket exhaust velocity, Ve.

You can get if from

T = dm/dt * Ve
where dm/dt is the rate of mass release, 12.7*10^-3 kg/1.90 s,
and T is the thrust in Newtons

To find the final velocity of the rocket, we can use the principle of conservation of momentum. The momentum before the rocket is fired is equal to the momentum after firing.

First, let's find the initial momentum of the system (the rocket plus rocket engine) before firing. The initial momentum is given by the product of the initial mass and the initial velocity of the system. Since the rocket is at rest initially, the initial velocity is 0.

Initial momentum = (initial mass of rocket + initial mass of rocket engine) * initial velocity
Initial momentum = (57.5 g + 25.5 g) * 0 = 0

Now, let's find the final momentum of the system after firing. The final momentum is given by the product of the final mass and the final velocity of the system. Let's find the final mass first.

The final mass is equal to the initial mass plus the mass of the fuel burned. Since the fuel mass is given in grams, we need to convert it to kilograms by dividing it by 1000.

Final mass = initial mass + fuel mass
Final mass = 57.5 g + 12.7 g = 70.2 g = 0.0702 kg

Now, we need to find the change in velocity, which is the change in momentum divided by the final mass.

Change in velocity = change in momentum / final mass

The change in momentum is equal to the thrust of the rocket engine multiplied by the duration of the burn.

Change in momentum = thrust * duration of burn
Change in momentum = 5.26 N * 1.90 s = 9.974 N·s

Change in velocity = 9.974 N·s / 0.0702 kg = 142.11 m/s

Finally, we can find the final velocity of the rocket by adding the change in velocity to the initial velocity (which is 0).

Final velocity = initial velocity + change in velocity
Final velocity = 0 + 142.11 m/s = 142.11 m/s

Therefore, the final velocity of the rocket is 142.11 m/s.