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Rocket Propulsion (Example of Variable Mass Situation)The principle of conservation of momentum can be applied to study the motion of a rocket. This application is of special significance as rocket is a system in which mass varies with time. v+dv Consider a rocket (may be a multistage rocket) moving vertically upwards from the surface of earth. Suppose that at time
According to the principle of conservation of momentum, the linear momentum of mass The velocity The factor In the Eq.(1.12)
Therefore, the Eq/(1.12) becomes The negative sign indicates that the velocity of the burnt gases w.r.t. the rocket i.e. Expression for velocity of rocket at any instant. From equation (1.13), we have Now, when Usually the exhaust velocity of the burnt gases is assumed to be constant throught the firing of the rocket. Therefore, or or or
Therefore, The Eq. (1.15) gives the velocity of the rocket at any time If at
Thus, velocity of the rocket at any instant is directly proportional to (i) the exhaust speed of the burnt gases and (ii) the natural logarithm of the ratio of the initial mass of the rocket to its mass at that instant. Thrust on the rocket. Dividing both sides of the equation (1.13) by Since Since the velocity of the burnt gases w.r.t. the rocket i.e. Here, Thus, thrust on the rocket at any instant is equal to the product of the exhaust speed of the burnt gases and the rate of combustion of the fuel at that instant. Burnt out speed of the rocket. The speed acquired by the rocket, when whole of the fuel gets burnt, is called the burnt out speed and it is the maximum speed that can be acquired by the rocket. Therefore, when the rocket acquires the burnt-out speed ( when In the equation (1.15), setting the above condition, we have
EXAMPLE 8—13 In the first second of its flight, a rocket ejects SOLUTION We have
EXAMPLE 8—14 Suppose the ratio of initial mass SOLUTION From Eq. (8-35), At the start of the flight, when the velocity of the rocket is zero, the ejected gases are moving downward, relative to the earth, with a velocity equal to the relative velocity
Example 1.23. Fuel is consumed at the rate of 100 kg/s in a rocket. The exhaust gases are ejected at a speed of Sol. Here, rate of consumption of the fuel,
speed with which the exhaust gases are ejected,
Now, thrust experienced by the rocket,
Also,
Exercises What is inertia ? Why do we call the Newton's first law as the law of inertia ? Explain.
5, State and explain Newton's second law of motion. How does this law give a unit (measure) of force ? Define SI unit of force. State Newton's second law of motion and prove that impulse is equal to the change in momentum produced.
State Newton's second law of motion and prove that impulse is equal to the change in momentum.
12. State the principle of conservation of linear momentum. Explain, how you will prove this law. Explain one example, where we make use of this law. Type E. On Rocket propulsion 21. A rocket consumes 24 kg of fuel per second The burnt gases escape the rocket at a speed of 6.4 km/s relative to the rocket. Calculate the upthrust received by the rocket. Also calculate the velocity acquired by the rocket, when its mass reduces to 1/100 of its initial mass. [Ans. 1.536 x 105 N; 2.94 x 105 m/s] 22. A rocket of initial mass 6,000 kg ejects mass at a constant rate of 16 kg/s with constant relative speed of 11 km/s. What is the acceleration of the rocket a minute after the blast ? (Neglect gravity) [Ans.34.92 m/s2 ]
A jet engine works on the principle of conservation of (A) mass. (B) energy. (C) linear momentum. (D) angular momentum.
Which of the following works on the principle of conservation of linear momentum ? (A) jet (B) aeroplane (C) rocket (D) all of these
If the force on a rocket moving with a velocity of 300 m/s is 210 N, then the rate of combustion of the fuel is (A) 07 kg/s (B) 14 kg/s (C) 0.07 kg/s (D) 10.7 kg/s
A 5000 kg rocket is set for vertical firing. The exhaust speed is 800 m/s. To give an initial upward acceleration of 20 m/s2, the amount of gas ejected per second to supply the needed thrust will be (g = 10 m s-2)
Date: 2015-01-12; view: 3692
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