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Relative Velocity in 2D

Two particles...

Question

Two particles move with constant velocities ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ . At the initial moment, the position vectors are ${\vec{r}}_{1}$ and ${\vec{r}}_{2}$ respectively. The condition for the collision of the particles is

A

${\vec{r}}_{1} - {\vec{r}}_{2} = {\vec{v}}_{1} - {\vec{v}}_{2}$

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B

$\frac{{\vec{r}}_{1} - {\vec{r}}_{2}}{|{\vec{r}}_{1} - {\vec{r}}_{2}|} = \frac{{\vec{v}}_{2} - {\vec{v}}_{1}}{|{\vec{v}}_{2} - {\vec{v}}_{1}|}$

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C

${\vec{r}}_{1} . {\vec{v}}_{1} = {\vec{r}}_{2} . {\vec{v}}_{2}$

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D

${\vec{r}}_{1} \times {\vec{v}}_{1} = {\vec{r}}_{2} \times {\vec{v}}_{2}$

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Solution

The correct option is B
$\frac{{\vec{r}}_{1} - {\vec{r}}_{2}}{|{\vec{r}}_{1} - {\vec{r}}_{2}|} = \frac{{\vec{v}}_{2} - {\vec{v}}_{1}}{|{\vec{v}}_{2} - {\vec{v}}_{1}|}$

Step 1: The position vector at the time of the collision:
For the collision of the two particles, the final position of the two particles should be the same at the same time.
The time required to reach the final position can be determined by dividing the change in position by the change in velocity.
Since the particles are moving with constant velocities, the time required to reach the final position can be expressed as follows:
$t = \frac{|{\vec{r}}_{1} - {\vec{r}}_{2}|}{|{\vec{v}}_{2} - {\vec{v}}_{1}|}$
Here, $t$ is the time required to reach the final position.
Step 2: Calculation of the final position of particle one.
The total displacement of particle one is the multiplication of the velocity of particle one to the time required.
Hence the final position of particle one can be determined by adding the initial position with the total displacement.
${\vec{r}}_{1} + {\vec{v}}_{1} t$
Step 3: Calculation of the final position of particle two.
Similarly, the final position of particle two can also be determined as follows:
${\vec{r}}_{2} + {\vec{v}}_{2} t$
Step 4: The condition for the collision
At the time of the collision, the positions of the particles will be the same.
${\vec{r}}_{1} + {\vec{v}}_{1} t = {\vec{r}}_{2} + {\vec{v}}_{2} t {\vec{r}}_{1} - {\vec{r}}_{2} = ({\vec{v}}_{2} - {\vec{v}}_{1}) t {\vec{r}}_{1} - {\vec{r}}_{2} = ({\vec{v}}_{2} - {\vec{v}}_{1}) \frac{|{\vec{r}}_{1} - {\vec{r}}_{2}|}{|{\vec{v}}_{2} - {\vec{v}}_{1}|} \frac{{\vec{r}}_{1} - {\vec{r}}_{2}}{|{\vec{r}}_{1} - {\vec{r}}_{2}|} = \frac{{\vec{v}}_{2} - {\vec{v}}_{1}}{|{\vec{v}}_{2} - {\vec{v}}_{1}|}$
Hence, option (B) is correct.

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