4. We should like to make a rocket go as fast as possible, but what would be rea

Question

4. We should like to make a rocket go as fast as possible, but what would be realistic fora final speed? Assume that a rocket starts at rest and in space so that you can neglect gravity. Imagine that the exhaust speed is w 2,000 ms relative to the rocket. a. What fraction of the initial mass of the rocket and fuel is not fuel, if the final speed is to be 3,000 ms-1? 1.00×10-3 c where c is the speed of light? kg. What would be the initial mass of the rocket needed in part b it this were to be the b. What fraction of the initial mass of the rocket and fuel is not fuel, if the final speed is to be c. The Apollo 11 command and service module used by the astronauts had a mass of 28,801 mass that was not the fuel?

Explanation / Answer

We will use the concept of conservation of momentum to solve these problems

We know that conservation of momentum states that that the total momentum of an isolated body is always constant if no outer force is applied on it.

In the given case the rocket is in space and no gravitational force is applying on it, So the rocket is isolated and we can apply the conservation of momentum.

Inititally the rocket with its fuel is at rest so the initial momentum is zero and from conservation of momentum final momentum is also zero.

a) Let the mass of rocket without fuel be 'M'

Let the mass of fuel be 'm'

So the total mass is M + m

Given that the exhaust speed i.e velocity of burning fuel in opposite direction of rocket is 2000m/s

Also given that the final speed of only rocket without fuel is 3000m/s

So final momentum is 3000*M - 2000*m

minus sign is there because the fuel exhaust speed is opposite in direction.

Now since momentum is 0 so, 3000*M - 2000*m = 0

which gives M = 2m/3

We have to find out the fraction of the initial mass of the rocket and fuel which is not fuel i.e M/(M+m)

which is '2/5' by substituting M = 2m/3

b) Similarly as above we can write 10^(-3)c *M - 2000*m = 0 where c = 3 * 10^8

so 3*10^5 *M - 2000*m = 0 which gives M = m/150

So M/(M+m) = 1/151

c) Given the mass of rocket which is not fuel i.e M = 28801 kg

And from above we know that M = m/150 so m = 150M = 150 * 28801 = 4320150 kg

We have to find the initial mass i.e M + m = 28801 + 4320150 = 4348951 kg

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