When designing rockets to transport a given payload into space, rocket scientists must determine the optimal mass of fuel for the rocket to carry. How can they determine the maximum velocity a rocket will attain based on the amount of fuel it carries?

What is known:

• The total initial mass of a rocket and its fuel is m_o.
• The fuel has an initial mass m_f, and burns at a constant rate of dm_f/dt = c.
• The fuel is expelled from the rocket at a constant speed v_f relative to the rocket.
• The rocket is fired vertically from rest.
• Drag resistance of the air may be neglected.

What is the velocity of the rocket in terms of given parameters once it has burned all of its fuel?

• Use conservation of linear momentum to determine the force exerted on an object as a result of emitting a continuous flow of mass.
• Use Newton's Second Law to derive an equation relating the sum of forces acting on the rocket to the rocket's acceleration.
• Integrate to determine the rocket's velocity.