Two cars moving parallelly to each other with the velocities shown. Initial separation between cars is 𝑟 = 10 m. Find angular momentum of car A w.r.t. car B.

Correct Answer: 100m

Analysis of the Given Image:
From the provided diagram, we can observe two cars moving along parallel straight tracks:

- Mass of Car A, $m_1=m$
- Mass of Car B, $m_2=\frac{3}{2}m$
- Velocity of Car A, $v_A=36\text{km/h}$
- Velocity of Car B, $v_B=72\text{km/h}$
- Perpendicular distance (initial separation) between their lines of motion, $r=10\text{m}$

Step 1: Convert velocities from km/h to m/s

To convert velocity from km/h to m/s, multiply by $\frac{5}{18}$:

$v_A=36\×\frac{5}{18}=10\text{m/s}$

$v_B=72\×\frac{5}{18}=20\text{m/s}$

Step 2: Find the relative velocity of Car A with respect to Car B

Since both cars are moving in the same direction along parallel paths:

$v_{A/B}=v_A-v_B=10\text{m/s}-20\text{m/s}=-10\text{m/s}$

The magnitude of the relative velocity is:

$|<!--\; |\; -->v_{A/B}|<!--\; |\; -->=10\text{m/s}$

Step 3: Calculate the angular momentum of Car A with respect to Car B

The angular momentum $L$ of Car A with respect to Car B is given by the formula:

$L=m_1\&times;|<!--\; |\; -->v_{A/B}|<!--\; |\; -->\&times;r_{\&perp;}$

where $r_{\&perp;}=r=10\text{m}$ is the perpendicular distance between the lines of motion of the two cars.

Substituting the given values:

$L=m\&times;10\&times;10=100m\text{kg}\&middot;{\text{m}}^2\text{/s}$

Conclusion:
The angular momentum of car A w.r.t. car B is 100m.