Three particles are projected simultaneously in the same vertical plane with different velocity at different angles. At later time t, the area of the triangle formed by the particles is proportional to a) t b) t2 c)t3 d) t4

To determine how the area of the triangle formed by three particles projected in the same vertical plane relates to time, we need to consider the motion of each particle and how their positions change over time. The key lies in understanding the geometric properties of the triangle formed by these particles as they move.

Understanding Particle Motion

When particles are projected at different angles and with different velocities, their positions at any time \( t \) can be described using the equations of motion. For each particle, the horizontal and vertical positions can be expressed as:

• Horizontal position: \( x_i = v_{i} \cdot \cos(\theta_i) \cdot t \)
• Vertical position: \( y_i = v_{i} \cdot \sin(\theta_i) \cdot t - \frac{1}{2} g t^2 \)

Here, \( v_i \) is the initial velocity of the \( i \)-th particle, \( \theta_i \) is the angle of projection, and \( g \) is the acceleration due to gravity. The positions of the three particles will vary based on their respective velocities and angles.

Area of the Triangle

The area \( A \) of a triangle formed by three points in a plane can be calculated using the determinant formula:

\( A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|

Substituting the expressions for \( x_i \) and \( y_i \) into this formula will yield an expression for the area in terms of time \( t \).

Proportionality to Time

As we analyze the resulting expression, we find that the area \( A \) will depend on the square of the time \( t \). This is because the positions of the particles are linear functions of time (for both \( x \) and \( y \)), and the area of a triangle formed by these linear functions will yield a quadratic relationship with respect to time.

Thus, the area of the triangle formed by the three particles is proportional to \( t^2 \). Therefore, the correct answer to the question is:

• b) t²

Visualizing the Concept

To visualize this, imagine that as time progresses, the particles move away from each other, effectively increasing the area of the triangle they form. The rate at which this area increases is not linear but quadratic, reflecting the fact that both the horizontal and vertical distances increase with time. This is similar to how the area of a square increases with the square of its side length.

In summary, the area of the triangle formed by the three particles projected at different angles and velocities increases with the square of time, leading us to conclude that the area is proportional to \( t^2 \). This relationship highlights the fascinating interplay between geometry and kinematics in physics.