You are the ground-control commander of a 2000 kg scientific rocket that is appr

Question

You are the ground-control commander of a 2000 kg scientific rocket that is approaching Mars at a speed of 2.9×104 km/hr. It needs to quickly slow to 2.0×104 km/hr to begin a controlled descent to the surface. If the rocket enters the Martian atmosphere too fast it will burn up, and if it enters too slowly, it will use up its maneuvering fuel before reaching the surface and will crash. The rocket has a new braking system: Several 5.0 kg "bullets" on the front of the rocket can be fired straight ahead. Each has a high-explosive charge that fires it at a speed of 111,000 m/s relative to the rocket.

You need to send the rocket an instruction to tell it how many bullets to fire. Success will bring you fame and glory, but failure of this $500,000,000 mission will ruin your career.

Explanation / Answer

You are the ground-control commander of a 2000 kg scientific rocket that is approaching Mars at a speed of 2.9×104 km/hr. It needs to quickly slow to 2.0×104 km/hr to begin a controlled descent to the surface. If the rocket enters the Martian atmosphere too fast it will burn up, and if it enters too slowly, it will use up its maneuvering fuel before reaching the surface and will crash. The rocket has a new braking system: Several 5.0 kg "bullets" on the front of the rocket can be fired straight ahead. Each has a high-explosive charge that fires it at a speed of 111,000 m/s relative to the rocket.

You need to send the rocket an instruction to tell it how many bullets to fire. Success will bring you fame and glory, but failure of this $500,000,000 mission will ruin your career.
   According to law of conservation of linear momentum    net momentum before firing   =   net momentum after firing    let 'n' bullets mus be fired, then   M * U   =   mr * vr   +   n * mb * vb    Mass of rocket + bullets   =   2000   kg    U   =   initial velocity of rocket   =   2.9 * 104   km/hr          =   2.9 * 104 * 5 / 18   =   8055.56   m/s    mr   =   mass of rocket alone after firing n bullets          =  ( 2000   -   n * 5 )   kg    vr   =   final velocity of rocket          =   2.0 * 104   km/hr          =   2.0 * 104 * 5 / 18          =   5555.56   m/s    mb   =   mass of each bullet   =   5   kg    vb   =   velocity of bullet in ground frame of reference          =   111000   +   vr          =   111000   +   5555.56          =   116555.56   m/s    =>   2000 * 8055.56   =   (2000 - 5 * n) * 5555.56   +   5 * n * 116555.56    1.611 * 107   =   1.111 * 107   -   2.778 * 104 * n   +   5.828 * 105 * n    n   =   (1.611 - 1.111) * 107 / ( 58.28 - 2.778) * 104          =   500 / 55.50          =   9.008   ˜   9       Hence 9   bullets must be fired.          =   500 / 55.50          =   9.008   ˜   9       Hence 9   bullets must be fired.

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