An object at rest explodes into three fragments. FIGURE EX11.32 shows the momentum vectors of two of the fragments. What is the momentum of the third fragment? Write your answer using unit vectors.

A particle exploded into three fragments in which momentum are given for two fragment and we need to find the momentum of thrid one.

When two things collide and cling together and move as one, the collision is inelastic, meaning that the momentum is preserved but the energy is not. Both objects have the same ultimate velocity, thus we utilize the conservation of momentum law, which is expressed in the form of the equation

${\overrightarrow{P}}_i={\overrightarrow{P}}_f$

The item fractures into three pieces, each of which is initially at rest, implying that the initial momentum is zero.

To fulfill the conservation law of momentum, the end momentum should be the same. Therefore, the equation will be

$$\begin{gathered} {\overrightarrow{P}}_f=0 \\ {\overrightarrow{p}}_{f1}+{\overrightarrow{p}}_{f2}+{\overrightarrow{p}}_{f3}=0 \\ {\overrightarrow{p}}_{f3}=-{\overrightarrow{p}}_{f1}-{\overrightarrow{p}}_{f2} \end{gathered}$$

From the figure, the final momentum of fragment 1 ${\left({\overrightarrow{p}}_f\right)}_1=(-2\hat{i}+2\hat{j})kg·m/s$and

the final momentum of fragment 2, ${\left({\overrightarrow{p}}_f\right)}_2=(3\hat{i}+0\hat{j})kg·m/s$

Substituting,

role="math" localid="1647797494965" $$\begin{gathered} {\overrightarrow{p}}_{f3}=-(-2i+2j)-(3i+0j) \\ {\overrightarrow{p}}_{f3}=(-i-2j)kg.m/s \end{gathered}$$