Chapter 6, Problem 1WUE

Math Review Solve the two equations mυ_i + MV_i = mυ_f + MV_f and υ_i − V_i = − (υ_f − V_f) for (a) υ_f and (b) V_f if m = 2.00 kg, υ_i = 4.00 m/s, M = 3.00 kg, and V_i = 0. (See Section 6.3.)

To determine

Solving of two given equations $m{v}_i+M{V}_i=m{v}_f+M{V}_f$ and $v_i-V_i=-\left(v_f-V_f\right)$ for $v_f$.

Answer to Problem 1WUE

Solution: The final velocity $v_f$ of mass $m$ is $-0.800\text{m/s}$.

Explanation of Solution

Given info: Masses $m$ and $M$ are $2.00\text{kg}$ and $3.00\text{kg}$, the initial velocity of mass $m$ is $4.00\text{m/s}$, and the initial velocity of mass $M$ is $0\text{m/s}$.

The given two equations are,

$m{v}_i+M{V}_i=m{v}_f+M{V}_f$      (I)

$v_i-V_i=-\left(v_f-V_f\right)$.

• $m$ and $M$ are masses.
• $v_i$ is initial velocity of mass $m$.
• $V_i$ is final velocity of mass $M$.

Rearrange the second equation to get $V_i$

$V_f=v_f-V_i+v_i$      (II)

Substitute equation (II) in (I) and re-arrange.

$\begin{array}{l}m{v}_i+M{V}_i=m{v}_f+M\left(v_f-V_i+v_i\right)\\ m{v}_i+M{V}_i=m{v}_f+M{v}_f-M{V}_i+M{v}_i\\ \left(m-M\right)v_i+2M{V}_i=\left(m+M\right)v_f\\ v_f=\frac{\left(m-M\right)v_i+2M{V}_i}{\left(m+M\right)}\end{array}$

The formula for the final velocity of mass $m$ is,

$v_f=\frac{\left(m-M\right)v_i+2M{V}_i}{\left(m+M\right)}$

Substitute $2.00\text{kg}$ for $m$, $3.00\text{kg}$ for $M$, $4.00\text{m/s}$ for $v_i$, and $0\text{m/s}$ for $V_i$ to find $v_f$.

$\begin{array}{c}{v}_f=\frac{\left(2.00\text{kg}-3.00\text{kg}\right)\left(4.00\text{m/s}\right)+2\left(3.00\text{kg}\right)\left(0\text{m/s}\right)}{\left(2.00\text{kg}+3.00\text{kg}\right)}\\ =-0.800\text{m/s}\end{array}$

Thus, the value of $v_f$ is $-0.800\text{m/s}$.

Conclusion:

Therefore, the final velocity $v_f$ of mass $m$ is $-0.800\text{m/s}$.

To determine

Solving of two given equations $m{v}_i+M{V}_i=m{v}_f+M{V}_f$ and $v_i-V_i=-\left(v_f-V_f\right)$ for $V_f$.

Answer to Problem 1WUE

Solution: The final velocity $V_f$ of mass $M$ is $\text{3}\text{.20}\text{m/s}$.

Explanation of Solution

Given info: The initial velocity of mass $m$ is $4.00\text{m/s}$, and the initial velocity of mass $M$ is $0\text{m/s}$, and final velocity of mass $m$ is $-0.800\text{m/s}$.

The formula for the final velocity of mass $m$ is,

$V_f=v_f-V_i+v_i$

• $v_i$ is initial velocity of mass $m$.
• $V_i$ is final velocity of mass $M$.
• $v_f$ is final velocity of mass $m$

Substitute $4.00\text{m/s}$ for $v_i$, $0\text{m/s}$ for $V_i$, and $-0.800\text{m/s}$ for $v_f$ to find $V_f$.

$\begin{array}{c}{V}_f=-0.800\text{m/s}-0\text{m/s}+4.00\text{m/s}\\ =\text{3}\text{.20}\text{m/s}\end{array}$

Conclusion:

Therefore, the final velocity $V_f$ of mass $M$ is $\text{3}\text{.20}\text{m/s}$.