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Question: A rocket of mass 1000kg exhausts gases at a rate of \(4kg{{s}^{-1}}\) with a velocity \(3000m{{s}^{-...

A rocket of mass 1000kg exhausts gases at a rate of 4kgs−14kg{{s}^{-1}} with a velocity 3000ms−13000m{{s}^{-1}}. The thrust developed on the rocket is

Explanation

Solution

Recall the expression for thrust developed in the rocket as a resultant of exhaustion of gases in the opposite direction. All the required quantities are directly given in their SI units in the question. All you have to do is substitute the values accordingly and get the value of thrust in Newton.

Formula used:
Expression for thrust,
T=vdmdtT=v\dfrac{dm}{dt}

Complete step-by-step answer:
In the question the quantity that we are asked to find is the thrust of a rocket. Before going into the question, let us understand what is thrust. We could describe thrust quantitatively by Newton’s third law. When a system under consideration expels or accelerates mass in one direction, the accelerated mass will cause a force of equal magnitude and different direction on that system. The force that is applied on a surface in a direction normal to the surface is also thrust. Thrust has the same SI unit as that of force, that is, Newton (N).
Mathematically, the thrust is given by the expression,
T=vdmdtT=v\dfrac{dm}{dt} …………………………………. (1)
Where, T is the thrust generated, dmdt\dfrac{dm}{dt} is the rate of change of mass with respect to time and v is the velocity of exhaust gases measured relative to the rocket.
We are given the mass of the rocket as,
m=1000kgm=1000kg
Exhaustion rate of gas, that is, the rate change of mass with respect to time,
dmdt=4kgs−1\dfrac{dm}{dt}=4kg{{s}^{-1}}
Velocity of exhaust gases relative to rocket,
v=3000ms−1v=3000m{{s}^{-1}}
Substituting all the given values in (1) we get,
T=(3000ms−1)×(4kgs−1)T=\left( 3000m{{s}^{-1}} \right)\times \left( 4kg{{s}^{-1}} \right)
T=12000NT=12000N
Hence, the thrust developed on the rocket is 12000N.

Note: For the vertical launch of a rocket, the initial thrust acting on the rocket must be more than the weight of the rocket. Also, the thrust so developed is in the direction opposite to the direction of escaping gas. Thrust is also generated in a motor boat when the propellers are turned to accelerate water and as a resultant the boat is pushed in the opposite direction.