Explanation Given: The initial total mass of the rocket including fuel is m 0 . Mass flow ejected on firing is m ˙ e . Velocity of the ejection is v e . Cross sectional area at the nozzle is A e . Pressure at the nozzle exit is p e . Drag force acting on the rocket is F D = c t . Calculations: Consider the entire rocket and its contents which accelerate upward as the control volume. Consider steady flow of an ideal fluid relative to the control volume. Express the mass of the rocket as a function of time. m = m 0 − m ˙ e t Similarly, express the weight of the rocket ( W ) as a function of time. W = m g W = ( m 0 − m ˙ e t ) g The gage pressure force on the nozzle ( F e ) can be determined as follows: F e = p e A e Draw the free body diagram for the control volume considered as in Figure (1). Express the relation for mass flow ejected from the rocket when fired. m ˙ e = ρ e V e A e Write the equilibrium force equation in vertical direction for the control volume of Figure (1). Σ F = m d V cv d t + ∂ ∂ t ∫ cv V f / c v ρ a d V + ∫ c 0 c 0 V f / c v ( ρ a V f / cs ⋅ d A ) ∑ F y = ( m 0 − m ˙ e t ) d V d t + 0 + ( − V e ) ( ρ e V e A e ) F e − F D − W = ( m 0 − m ˙ e t ) d V d t + 0 + ( − V e ) ( m ˙ e ) p e A e − c t − ( m 0 − m ˙ e t ) g = ( m 0 − m e t ) d V d t + 0 + ( − V e ) ( m ˙ e ) ( m 0 − m e t ) d V d t = p e A e − c t − ( m 0 − m ˙ e t ) g + ( V e ) ( m ˙ e ) d V d t = p e A e ( m 0 − m e t ) − c t ( m 0 − m e t ) + ( V e ) ( m ˙ e ) ( m 0 − m e t ) − g Integrate the expression for V = 0 at t = 0

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ISBN: 9780134626055

Author: HIBBELER

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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Chapter 6, Problem 89P

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The velocity of the rocket when its gravitational acceleration is considered constant.

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Chapter 6, Problem 88P

Chapter 6, Problem 90P

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