Concept explainers
A rocket weighs 2600 lb, including 2200 lb of fuel, which is consumed at the rate of 25 lb/s and ejected with a relative velocity of 13,000 ft/s. Knowing that the rocket is fired vertically from the ground, determine (a) its acceleration as it is fired, (b) its acceleration as the last particle of fuel is being consumed, (c) the altitude at which all the fuel has been consumed, (d) the velocity of the rocket at that time.
(a)
The acceleration of the rocket as it is fired.
Answer to Problem 14.100P
The acceleration of the rocket as it is fired is
Explanation of Solution
Given information:
The fuel consumed rate is
The relative velocity is
The gross weight of the rocket is
The weight of the fuel is
Calculation:
Consider the acceleration due to gravity
Calculate the thrust force
Substitute
Calculate the mass
Substitute
Calculate the acceleration
Substitute
Hence, acceleration of the rocket as it is fired is
(b)
The acceleration of the rocket as the last particle of fuel is being consumed.
Answer to Problem 14.100P
The acceleration of the rocket as the last particle of the fuel is being consumed is
Explanation of Solution
Given information:
The fuel consumed rate is
The relative velocity is
The gross weight of the rocket is
The weight of the fuel is
Calculation:
Refer to part (a).
The thrust force
Calculate the weight of the rocket
Substitute
Calculate the mass
Substitute
Calculate the acceleration
Substitute
Hence, acceleration of the rocket as the last particle of the fuel is being consumed is
(c)
The altitude at which all the fuel has been consumed
Answer to Problem 14.100P
The altitude of the rocket is
Explanation of Solution
Given information:
The fuel consumed rate is
The relative velocity is
The gross weight of the rocket is
The weight of the fuel is
Calculation:
Consider the weight of the fuel as
Calculate the time
Integrate both sides of the Equation.
Substitute
Refer to sample problem 14.8 in the Text book,
Calculate the velocity
Consider
Substitute
Calculate the altitude
Integrate both sides of the Equation with respect to t.
Substitute
Consider
Differentiate both sides of the Equation (4) as shown below.
Substitute z for
Substitute
Calculate the altitude
Substitute
Hence, the altitude of the rocket is
(d)
The velocity of the rocket.
Answer to Problem 14.100P
The velocity of the rocket is
Explanation of Solution
Given information:
The fuel consumed rate is
The relative velocity is
The gross weight of the rocket is
The weight of the fuel is
Calculation:
Calculate the velocity
Substitute
Therefore, the velocity of the rocket is
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