# Solved: Consider a single-stage rocket that is launched vertically upward as shown in

**Chapter 1, Problem 21**

(choose chapter or problem)

When the mass m of a body moving through a force field is variable, Newton’s second law of motion takes on the form: If the net force acting on a body is not zero, then the net force F is equal to the time rate of change of momentum of the body. That is,

\(F=\frac{d}{d t}(m v)\),

where mv is momentum. Use this formulation of Newton’s second law in Problems 21 and 22.

*Note that when m is constant, this is the same as F = ma.

Consider a single-stage rocket that is launched vertically upward as shown in the accompanying photo. Let m(t) denote the total mass of the rocket at time t (which is the sum of three masses: the constant mass of the payload, the constant mass of the vehicle, and the variable amount of fuel). Assume that the positive direction is upward, air resistance is proportional to the instantaneous velocity v of the rocket, and R is the upward thrust or force generated by the propulsion system. Use (17) to find a mathematical model for the velocity v(t) of the rocket.

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