Question

Two particles are projected simultaneously in the same vertical plane, from the same point, both with different speeds and at different angles with horizontal. The path followed by one, as seen by the other, is

• A. a vertical line
• B. a parabola
• C. a hyperbola
• D. a straight line making a constant angle ($\ne$ 90${}^{\circ }$) with horizontal

Answer 1

The correct option is D a straight line making a constant angle ($\ne$ 90${}^{\circ }$) with horizontal

Let ${\overrightarrow{u}}_1$ and ${\overrightarrow{u}}_2$ be the initial velocities of the two particles and ${\theta }_1$ and ${\theta }_2$ be their angles of projection with the horizontal.
The velocities of the two particles after time t are,
${\overrightarrow{v}}_1$= ($u_{1}cos{\theta }_1$)$\hat{i}$ +($u_{1}sin{\theta }_1$-gt)$\hat{j}$

${\overrightarrow{v}}_2$= ($u_{2}cos{\theta }_2$)$\hat{i}$ + ($u_{2}sin{\theta }_2$-gt)$\hat{j}$

${\overrightarrow{v}}_{12}$= ${\overrightarrow{v}}_1$ -${\overrightarrow{v}}_2$= ($u_{1}cos\theta -u_{2}cos{\theta }_2$)$\hat{i}$ +($u_{1}sin{\theta }_1$-gt - $u_{2}sin{\theta }_2$+ gt)$\hat{j}$

${\overrightarrow{v}}_{12}$= ${\overrightarrow{v}}_1$ -${\overrightarrow{v}}_2$= ($u_{1}cos\theta -u_{2}cos{\theta }_2$)$\hat{i}$ +($u_{1}sin{\theta }_1$ - $u_{2}sin{\theta }_2$)$\hat{j}$

which is a constant. So the path followed by one, as seen by the other is straight line, making a constant angle with the horizontal.