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1. Consider a gun of mass M (when unloaded) that fires a shell of mass m with muzzle speed v. (That is, the shells speed relative to the gun is v.) Assuming that the gun is completely free to recoil (no external forces on gun or shell), use conservation of momentum to show that the shells speed relative to the ground is v/(1+m/M)
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Answer #1

Since you have asked about using F=dp/dt, I want to highlight one thing. During any process, if the net force acting on a system is zero, the exchange of momentum takes place forever as long as there is no any extra force that comes into play. So momentum conservation is always happening. But we are interested in the momentum before firing and after firing which is a result of the point I have made in the newly uploaded solution. The modification is provided in the box. Please check.

First we start by considering the bullet as well as gun as a single system. The net force acting on this system is zero. From Newtons second law we know that dp dt F- 0 p constant This says that the momentum remains constant for this system. While firing a bullet, the momentum of the system (bullet + gun) remains same before and after. This is called as conservation of linear momentum According to conservation of momentum, when the net force acting on the system is zero, the linear momentum of the system is equal before and after the collision. The initial momentum of the system is zero. If the recoil velocity of the gun relative to the ground is V and the velocity of the bullet is Vrelative to the ground. The final momentum is given by From conservation of linear momentum, since initial momentum is zero 1 These velocities are related to the velocity of the bullet relative to the shell as We substitute the above result in this to obtain V, the velocity of the shell relative to ground 7m We take 1to the right hand side. This gives us

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