In a rocket, the mass of the fuel is 90℅ of the total mass. The exhaust gasses are ejected at a speed of 2km/s. Find the maximum speed attained by the rocket with diagram. Neglect gravity and air resistance and take the initial speed of the rocket to be zero. (Given, log e 10 = 2.3)

Asked by rrpapatel | 18th Oct, 2018, 04:13: PM

Expert Answer:

At a given instant, let m be the total mass of rocket+fuel and u be the velocity.
Once small amount dm of fuel is exhausted its velocity increases by u+du. Let ue be the exhaust velocity.
 
Initial momentum = m×u
final momentum = (m - dm)×(u+du) - dm ×(ue-u) = m×u + m×du - dm×ue (term  dm×du is neglected)
 
Since gravity and air resistance are neglected, there in sno external force, hence momentum is conserved.

By momentum conservation, we have,  m×u + m×du - dm×ue = m×u ..............(1)
 
after simplification of eqn.(1), we get,  du = ue × (dm/m) ..............(2)
By integrating eqn.(2), we get

begin mathsize 12px style integral subscript u subscript i end subscript superscript u subscript f end superscript d u space equals space u subscript e integral subscript m subscript i end subscript superscript m subscript f end superscript fraction numerator d m over denominator m end fraction
u subscript f space minus space u subscript i space equals space u subscript e cross times ln m subscript f over m subscript i space........................ left parenthesis 3 right parenthesis end style

 
It is given that initial velocity ui = 0. Since 90% of initial mass is fuel and final mass is only rocket weight,  mi = 10×mf
 
Then we have from eqn.(3)  final maximum velocity uf = ue×ln(0.1) = -2×(-ln10) = 2 × 2.3 = 4.6 km/s

Answered by Thiyagarajan K | 19th Oct, 2018, 02:46: AM

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