An astronaut in space cannot use conventional means, such as a scale or balance, to determine the mass of an object. But she does have devices to measure distance and time accurately. She knows her own mass is $\mathbf{78}\mathbf{.}\mathbf{4}\mathbf{\text{kg}}$, but she is unsure of the mass of a large gas canister in the airless rocket. When this canister is approaching her at role="math" localid="1660235198768" $\mathbf{3}\mathbf{.}\mathbf{50}\mathbf{}\raisebox{1ex}{${\displaystyle \text{m}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \text{s}}$}\right.$, she pushes against it, which slows it down to$\mathbf{1}\mathbf{.}\mathbf{20}\mathbf{}\raisebox{1ex}{${\displaystyle \text{m}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \text{s}}$}\right.$ (but does not reverse it) and gives her a speed of $\mathbf{2}\mathbf{.}\mathbf{40}\mathbf{}\raisebox{1ex}{${\displaystyle \text{m}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \text{s}}$}\right.$. What is the mass of this canister?

Apply conservation of momentum to the astronaut and canister to find the mass of the canister.

According to the law of conservation of momentum, without the action of an external force, the combined momentum of two or more bodies operating on one another in an isolated system remains constant. As a result, momentum can neither be created nor destroyed.

Consider the given data as below.

The mass of the astronaut,$m_a=78.4\text{kg}$

Let the mass of the canister is $m_c$.

The initial velocity of the astronaut $v_{a1}=0$,

The initial velocity of the canister $v_{c1}=3.50\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.$,

The final velocity of the astronaut after collision,$v_{a2}=2.40\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.$

The final velocity of the canister after collision,$v_{c2}=1.20\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.$

The total initial momentum of the astronaut and canister is,

$p_1=m_a{v}_{a1}+m_c{v}_{c1}$ ..... (1)

The total final momentum of the astronaut and canister is,

$p_2=m_a{v}_{a2}+m_c{v}_{c2}$ ..... (2)

Apply the law of conservation to equations (1) and (2), you have

$p_1=p_2$

$$\begin{gathered} m_a{v}_{a1}+m_c{v}_{c1}=m_a{v}_{a2}+m_c{v}_{c2} \\ {m}_c{v}_{c1}-m_c{v}_{c2}=m_a{v}_{a2}-m_a{v}_{a1} \\ {m}_c\left(v_{c1}-v_{c2}\right)=m_a{v}_{a2}-m_a{v}_{a1} \\ {m}_c=\frac{m_a{v}_{a2}-m_a{v}_{a1}}{v_{c1}-v_{c2}} \end{gathered}$$

Substitute known values in the above equation.

role="math" localid="1660236077186" $\begin{array}{rcl}{m}_c& =& \frac{{\displaystyle \left(78.4\text{kg}\times 2.40\raisebox{1ex}{${\displaystyle \text{m}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \text{s}}$}\right.\right)-\left(78.4\text{kg}\times 0\right)}}{3.50{\displaystyle \raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.}\text{-}3.50{\displaystyle \raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.}}\\ & =& \frac{{\displaystyle 188.16\raisebox{1ex}{${\displaystyle \text{kg·m}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \text{s}}$}\right.\text{+0}}}{3.50{\displaystyle \raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.}}\\ & =& 81.8\text{kg}\end{array}$

Hence, the required mass of this canister is $8\begin{array}{rcl}& & 1\end{array}\begin{array}{rcl}& & .\end{array}\begin{array}{rcl}& & 8\end{array}\begin{array}{rcl}& & \text{kg}\end{array}$.