You are the ground control commander of a 2500 kg scientific rocket that is appr

Question

You are the ground control commander of a 2500 kg scientific rocket that is approaching Mars at a speed of 26491 km/hr. It needs to quickly slow to 15000 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 133000 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

Mass of Rocket = 2500 Kg
Speed of Rocket = 26491 km/hr = 7358.6 m/s
Mass of Bullet = 5.0 Kg
Speed of Bullet = 133000 m/s
Final speed of rocket = 15000 km/hr = 4166.7 m/s

Initial Momentum of Rocket = 2500 * 7358.6 Kgm/s
Initial Momentum of Rocket = 1.839 * 10^7 Kg m/s

Let n bullets are fired.
Momentum of Bullet = n*5.0 *133000
Momentum of Rocket = (2500 - n*5.0) * 4166.7 Kg m/s

Using Momentum Conservation -
1.839 * 10^7 = n*5.0 *133000 + (2500 - n*5.0) * 4166.7
Solving for n
n = 12.4

In whole number , n = 12 bullets need to fire.

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