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Difficulty: MediumMomentum and Collisions

A carpenter uses a 5 kg5\text{ kg} pneumatic nail gun. The gun fires a 50 gram50\text{ gram} nail forward with a velocity of 100 m/s100\text{ m/s}. Based on the conservation of momentum, if the gun is unrestrained, what is its backward recoil speed?

  1. A
    0.1 m/s0.1\text{ m/s}
  2. 1 m/s1\text{ m/s}Answer
  3. C
    100 m/s100\text{ m/s}
  4. D
    1,000 m/s1,000\text{ m/s}

Answer

The nail gun will have a backward recoil speed of 1 m/s1\text{ m/s}.
According to the law of conservation of momentum, the forward momentum of the nail must be perfectly balanced by the backward momentum of the nail gun. First, convert the nail's mass to standard units (50 g=0.05 kg50\text{ g} = 0.05\text{ kg}). The nail's momentum is 0.05 kg×100 m/s=5 kg⋅m/s0.05\text{ kg} \times 100\text{ m/s} = 5\text{ kg}\cdot\text{m/s}. Because the gun's mass is 5 kg5\text{ kg}, dividing the total momentum (55) by its mass (55) gives a recoil speed of 1 m/s1\text{ m/s}.

Step-by-Step Solution

1
Convert the nail's mass from grams to kilograms to ensure consistent standard units.
50 grams=0.05 kg50\text{ grams} = 0.05\text{ kg}.
Momentum equations require matching mass units on both sides, typically kilograms.
2
Calculate the forward momentum of the nail.
pnail=m×v=0.05 kg×100 m/s=5 kg⋅m/sp_{nail} = m \times v = 0.05\text{ kg} \times 100\text{ m/s} = 5\text{ kg}\cdot\text{m/s}.
Momentum is the product of mass and velocity.
3
Apply the principle of conservation of momentum.
Initial momentum is zero, so pgun=pnailp_{gun} = p_{nail}. The gun's backward momentum must be 5 kg⋅m/s5\text{ kg}\cdot\text{m/s}.
In a closed system with no external forces, the total momentum before and after firing must remain equal.
4
Calculate the backward recoil speed of the nail gun.
vgun=pgunmgun=5 kg⋅m/s5 kg=1 m/sv_{gun} = \frac{p_{gun}}{m_{gun}} = \frac{5\text{ kg}\cdot\text{m/s}}{5\text{ kg}} = 1\text{ m/s}.
Rearranging the momentum formula (p=mvp = mv) solves for the unknown velocity.

Key Concept

Conservation of Momentum (Recoil)
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Topics

Momentum and Collisions

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