Two particles $A$ and $B$, move with constant velocities $\vec{v}_1$ and $\vec{v}_2$ respectively. At the init

Question

Two particles $A$ and $B$, move with constant velocities $\vec{v}_1$ and $\vec{v}_2$ respectively. At the initial moment, their position vectors are $\vec{r}_1$ and $\vec{r}_2$ respectively. The condition for particles $A$ and $B$ for their collision will be:

Accepted Answer

1. **Condition for Collision:** For two particles to collide, they must occupy the same position at the same instant of time $t$ ($t > 0$).
2. **Position Equations:**
   - Position of particle A at time $t$: $\vec{r}_A(t) = \vec{r}_1 + \vec{v}_1 t$
   - Position of particle B at time $t$: $\vec{r}_B(t) = \vec{r}_2 + \vec{v}_2 t$
3. **Equating Positions:**
   $$ \vec{r}_1 + \vec{v}_1 t = \vec{r}_2 + \vec{v}_2 t $$
   $$ \vec{r}_1 - \vec{r}_2 = (\vec{v}_2 - \vec{v}_1) t $$
4. **Direction Analysis:** The equation implies that the relative displacement vector $(\vec{r}_1 - \vec{r}_2)$ is parallel to the relative velocity vector $(\vec{v}_2 - \vec{v}_1)$. Since time $t$ must be positive for a future collision, these two vectors must point in the same direction.
5. **Unit Vectors:** If two vectors are in the same direction, their unit vectors must be equal .
   $$ \hat{r}_{relative} = \hat{v}_{relative} $$
   $$ \frac{\vec{r}_1 - \vec{r}_2}{|\vec{r}_1 - \vec{r}_2|} = \frac{\vec{v}_2 - \vec{v}_1}{|\vec{v}_2 - \vec{v}_1|} $$

Step-by-Step Solution

Practice All PHYSICS Questions