Question

Two rocks collide with each other in outer space, far from all other objects. Rock 1 with mass 5 kg has velocity $〈\mathbf{30}\mathbf{,}\mathbf{45}\mathbf{,}\mathbf{-}\mathbf{20}〉$m/s before the collision and $〈\mathbf{-}\mathbf{10}\mathbf{,}\mathbf{50}\mathbf{,}\mathbf{-}\mathbf{5}〉$m/s after the collision. Rock 2 with mass 8 kg has velocity $\langle -9,5,4\rangle$m/s before the collision. Calculate the final velocity of rock 2.

Short answer

$\left(16,1.875,-5.375\right)\text{\hspace{0.33em}m/s}$

Step-by-step solution

Identification of the given data 

The given data can be listed below as,

• The mass of rock 1 is,$m_1=5\text{\hspace{0.33em}kg}$
• The velocity of rock 1 before colliding to rock 2 is,$u_1=〈30,45,-20〉\text{\hspace{0.33em}m/s}$
• The velocity of rock 1 after colliding to rock 2 is,$v_1=〈-10,50,-5〉\text{\hspace{0.33em}m/s}$
• The mass of rock 2 is,$m_2=8\text{\hspace{0.33em}kg}$
• The velocity of rock 2 before colliding to rock 2 is,$u_2=〈-9,5,4〉\text{\hspace{0.33em}m/s}$

Explanation of law of conservation of momentum

The total momentum experienced by a system before collision of two objects is same as that of the total momentum of the same system after the collision of two objects. It is expressed as follows,

${\mathbf{m}}_1{\mathbf{u}}_1\mathbf{+}{\mathbf{m}}_2{\mathbf{u}}_2\mathbf{=}{\mathbf{m}}_1{\mathbf{v}}_1\mathbf{+}{\mathbf{m}}_2{\mathbf{v}}_2$

Here, $m_1$is the mass of object 1,$u_1$ is the initial velocity of object 1,$v_1$ is the final velocity of object 1, $m_2$is the mass of object 2,$u_2$ is the initial velocity of object 2,$v_2$ is the final velocity of object 2.

Determination of the final velocity of rock 2.

Substitute all the values in the above expression.

$\begin{array}{c}5\left(30,45,-20\right)+8\left(-9,5,4\right)=5\left(-10,50,-5\right)+8v_2\\ 8v_2=5\left(30,45,-20\right)+8\left(-9,5,4\right)-5\left(-10,50,-5\right)\\ 8v_2=\left(128,15,-43\right)\\ v_2=\frac{\left(128,15,-43\right)}{8}\\ =\left(16,1.875,-5.375\right)\text{\hspace{0.33em}m/s}\end{array}$

Thus, the final velocity of rock 2 is .$\left(16,1.875,-5.375\right)\text{\hspace{0.33em}m/s}$