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Chapter 3: Problem 391

A bag of sand of mass \(\mathrm{m}\) is suspended by rope. a bullet of mass \((\mathrm{m} / 30)\) is fired at it with a velocity \(\mathrm{V}\) and gets embedded into it. The velocity of the bag finally is (A) \((31 \mathrm{~V} / 30)\) (B) \((30 \mathrm{~V} / 31)\) (C) \((\mathrm{V} / 31)\) (D) \((\mathrm{V} / 30)\)

Short Answer

Expert verified
The final velocity of the bag (\(V_f\)) after being hit by the bullet is: \(V_f = \frac{31V}{30}\). The correct answer is (A) \(\frac{31V}{30}\).

Step by step solution

01

Identify the initial momentum of the system

Before the collision, only the bullet is moving with a velocity of V, and the bag is stationary. So, the initial momentum of the system is the momentum of the bullet alone: Initial Momentum \(= m_{bullet} \times V\) Since the bullet mass is given as \(\frac{m}{30}\), we have: Initial Momentum \(= \frac{m}{30} \times V\)
02

Identify the final momentum of the system

After the collision, the bullet gets embedded into the bag, and they move together with the same final velocity, which we can denote as \(V_f\). The total mass of the system after the collision is \(\frac{31m}{30}\). Therefore, the final momentum of the system will be: Final Momentum \(= (m + \frac{m}{30}) \times V_f = \frac{31m}{30} \times V_f\)
03

The principle of conservation of linear momentum

According to the principle of conservation of linear momentum, the initial momentum of the system should equal the final momentum of the system. Therefore, we can set the initial momentum and final momentum equal to each other and solve for the final velocity, \(V_f\): \(\frac{m}{30} \times V = \frac{31m}{30} \times V_f\)
04

Solve for the final velocity, \(V_f\)

Divide both sides of the equation by \(\frac{m}{30}\) to solve for the final velocity: \(V_f = \frac{(\frac{31m}{30} \times V_f)}{(\frac{m}{30} \times V)}\) Now, cancel out the terms \(\frac{m}{30}\) on both sides: \(V_f = \frac{31}{30} \times V\) Thus, the final velocity of the bag is: \(V_f = \frac{31V}{30}\) Comparing this result with the given options, we find that the correct answer is: (A) \(\frac{31V}{30}\)

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Most popular questions from this chapter

A ball of mass \(0.2 \mathrm{~kg}\) is thrown vertically upwards by applying a force by hand. if the hand moves \(0.2 \mathrm{~m}\) while applying the force and the ball goes up to \(2 \mathrm{~m}\) height further. find the magnitude of the force. (Consider \(\mathrm{g}=10 \mathrm{~ms}^{-1}\) ) (A) \(16 \mathrm{~N}\) (B) \(20 \mathrm{~N}\) (C) \(22 \mathrm{~N}\) (D) \(4 \mathrm{~N}\)

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