Rocket problem: Newton's 2nd Law

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SUMMARY

The discussion focuses on solving a rocket problem based on Newton's 2nd Law, specifically analyzing the rocket's acceleration while neglecting air resistance. The key equations involve the mass of the rocket as a function of time, defined as m(t) = M1 + (M2 - bt), where M1 is the mass without fuel, M2 is the initial fuel mass, and b is the fuel burn rate in kg/s. Participants are guided to derive the burnout velocity and height at the moment the fuel is exhausted, emphasizing the application of separation of variables in solving the equations.

PREREQUISITES
  • Understanding of Newton's 2nd Law of Motion
  • Basic knowledge of calculus, particularly separation of variables
  • Familiarity with kinematic equations related to motion
  • Concept of rocket propulsion and mass flow rate
NEXT STEPS
  • Study the derivation of rocket equations using Newton's 2nd Law
  • Learn about the concept of burnout velocity in rocketry
  • Explore the application of separation of variables in differential equations
  • Investigate the effects of varying fuel burn rates on rocket performance
USEFUL FOR

Students studying physics, particularly those focusing on mechanics and dynamics, as well as educators looking for practical examples of Newton's laws in action.

hana.e.kim
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Homework Statement


In this problem we explore a slightly more realistic model of a rocket's acceleration still neglecting air resistance. Consider a rocket fired straight up from rest burning fuel at the constant rate of b kg/s. Let v=v(t) be the velocity of the rocket at time t and suppose that the velocity u of the exhaust gas is constant. Let m=m(t) be the mass of the rocket at time t, and note that m is not constant. From Newton's second law it can be shown that F=m(dv/dt) - uv where the force F= -mg and g is acceleration due to gravity, thus, m(dv/dt) - uv = -mg. Let M1 be the mass of the rocket without fuel and M2 be the initial mass of the fuel.

a. Find an equation for the mass m at time t in terms of M1, M2, and b.
b. Substitute this expression for m in equation one and solve the resulting equation for b. Use separation of variables.
c. Determine the velocity of the rocket at the time that the fuel is exhausted. This is called the burnout velocity.
d. Find the height of the rocket at the burnout time.


Homework Equations


See Newton's 2nd Law above.


The Attempt at a Solution


I honestly don't even know how to start this problem, so any help would be greatly appreciated. Thank you!
 
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Hi hana! :smile:
hana.e.kim said:
… I honestly don't even know how to start this problem, so any help would be greatly appreciated. Thank you!

ok, start with a. …

what is the mass m at time t in terms of M1, M2, and b ?
 

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