Solved problem: Conservation of linear momentum principle

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Statement

difficulty

A competitor shoots a plate by using a rifle of 2,5 Kg on a skeet shooting competition. Knowing that the ball is 23 g and it is shot horizontally at 350 m/s, what is the recoil of the rifle?

Solution

Data

mass of the rifle $m_{r} = 2.5 k g$	velocity before the shot ${\vec{v}}_{i r} = 0 m / s$	velocity of the rifle after the shot ${\vec{v}}_{f r} = ? m / s$
mass of the bullet $m_{b} = 23 \cdot 10^{- 3} = 0.023 k g$	velocity of the bullet before the shot ${\vec{v}}_{i b} = 0 m / s$	velocity of the bullet after the shot ${\vec{v}}_{f b} = 350 \cdot \vec{i} m / s$

Resolution

If we consider the bullet and the rifle as a system, the outer forces to it are cancel since, initially, both are held by the handle, therefore the conservation linear momentum principle, momentums before the shot ( ${\vec{p}}_{i}$ ) and after the shot ( ${\vec{p}}_{f}$ ) are the same:

${\vec{p}}_{i} = {\vec{p}}_{f}$

For this reason we are going to calculate the linear momentum of the system at that time, having into account the momentum of a system is equivalent to the sum of momentums of the bodies separatly:

Momentum before the shot	${\vec{p}}_{i} = {\vec{p}}_{i b} + {\vec{p}}_{i r} \Rightarrow {\vec{p}}_{i} = m_{b} \cdot {\vec{v}}_{i b} + m_{r} \cdot {\vec{v}}_{i r} \Rightarrow {\vec{p}}_{i} = 23 \cdot 10^{- 3} \cdot 0 + 2.5 \cdot 0 \Rightarrow {\vec{p}}_{i} = 0 k g \cdot m / s$
Momentum after the shot	${\vec{p}}_{f} = {\vec{p}}_{f b} + {\vec{p}}_{f r} \Rightarrow {\vec{p}}_{f} = m_{b} \cdot {\vec{v}}_{f b} + m_{r} \cdot {\vec{v}}_{f r} \Rightarrow {\vec{p}}_{f} = 23 \cdot 10^{- 3} \cdot 350 \cdot \vec{i} + 2.5 \cdot {\vec{v}}_{f r} \Rightarrow {\vec{p}}_{f} = 8.05 \cdot \vec{i} + 2.5 \cdot {\vec{v}}_{f r}$

Matching both linear momentums and solving for ${\vec{v}}_{f r}$ , we can calculate the rifle´s speed of recoil after the shot.

$0 = 8.05 \cdot \vec{i} + 2.5 \cdot \vec{v_{f r}} \Rightarrow \vec{v_{f r}} = \frac{- 8.05}{2.5} \cdot \vec{i} \Rightarrow \vec{v_{f r}} = - 3.22 \cdot \vec{i}$

Formulas worksheet

These are the main formulas that you must know to solve this exercise. If you are not clear about their meaning, we recommend you to check the theory in the corresponding sections. Furthermore, you will find in them, under the Formulas tab, the codes that will allow you to integrate these equations in external programs like Word or Mathematica.

Formulas

Related sections

\sum \vec{F} = 0 \Leftrightarrow \vec{p} = constant \Leftrightarrow \frac{d \vec{p}}{d t} = 0

Newton´s First Law

Momentum Conservation Principle

\vec{p} = m \cdot \vec{v}

Linear Momentum

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