Solved problem: Collision velocity and momentum conservation

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Statement

difficulty

A 3 Kg ball is moving towards left at 6 m/s. It collides with a 5 Kg ball moving towards right at 2 m/s. After the collision the second ball went through the left at 3 m/s. Calculate the velocity of the first ball after the crash.

Solution

Data

First´s ball mass m₁ = 3 kg
Second´s ball mass m₂ = 5 kg
First´s ball velocity before the crash ${\vec{v}}_{0_{1}} = - 6 \cdot \vec{i} m / s$
Second´s ball velocity before the crash ${\vec{v}}_{0_{2}} = 2 \cdot \vec{i} m / s$
Second´s ball velocity after the crash ${\vec{v}}_{f_{2}} = - 3 \cdot \vec{i} m / s$

Resolution

The linear momentum conservation principle establishes that, in absence of outer forces, the momentum is preserved. So, the linear momentum before the crash and after it must be the same.

$\sum \vec{F} = 0 \Rightarrow \vec{p} = constante \Rightarrow {\vec{p}}_{0} = {\vec{p}}_{f}$

The initial momentum of the system is made by the sum of momentums of each one of the balls:

${\vec{p}}_{0} = m_{1} \cdot {\vec{v}}_{0}_{1} + m_{2} \cdot {\vec{v}}_{0}_{2} = 3 \cdot (- 6 \cdot \vec{i}) + 5 \cdot (2 \cdot \vec{i}) = - 8 \cdot \vec{i} k g \cdot m / s$

The final momentum is given by:

${\vec{p}}_{f} = m_{1} \cdot {\vec{v}}_{f}_{1} + m_{2} \cdot {\vec{v}}_{f}_{2} = 3 \cdot {\vec{v}}_{f}_{1} + 5 \cdot (- 3 \cdot \vec{i})$

We are ready now to solve the final velocity of the first ball:

${\vec{p}}_{0} = {\vec{p}}_{f} \Rightarrow - 8 \cdot \vec{i} = 3 \cdot {\vec{v}}_{f}_{1} + 5 \cdot (- 3 \cdot \vec{i}) \Rightarrow 7 \cdot \vec{i} = 3 \cdot {\vec{v}}_{f}_{1} \Rightarrow {\vec{v}}_{f}_{1} = \frac{7}{3} \cdot \vec{i} m / s$

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.

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