Two particles are projected from the ground simultaneously with speed 20m/s and 9m/s at angle 30 and 60 with horizontal in the same direction. the maximum distance between them till both of them strike the ground is

To determine the maximum distance between the two particles projected at different speeds and angles, we need to analyze their trajectories. Let's break this down step by step.

Understanding the Motion of Each Particle

Both particles are projected from the ground, and their motion can be described using the equations of projectile motion. The key factors to consider are their initial speeds, angles of projection, and the time of flight.

Particle 1: Speed 20 m/s at 30 Degrees

• Initial Velocity Components:
  • Horizontal Component: \( V_{1x} = V_1 \cdot \cos(\theta_1) = 20 \cdot \cos(30^\circ) = 20 \cdot \frac{\sqrt{3}}{2} \approx 17.32 \, \text{m/s} \)
  • Vertical Component: \( V_{1y} = V_1 \cdot \sin(\theta_1) = 20 \cdot \sin(30^\circ) = 20 \cdot \frac{1}{2} = 10 \, \text{m/s} \)
• Time of Flight: The time of flight \( T_1 \) can be calculated using the formula \( T = \frac{2V_{y}}{g} \), where \( g \approx 9.81 \, \text{m/s}^2 \):
  • \( T_1 = \frac{2 \cdot 10}{9.81} \approx 2.04 \, \text{s} \)
• Horizontal Range: The horizontal distance covered by Particle 1 is given by \( R_1 = V_{1x} \cdot T_1 \):
  • \( R_1 = 17.32 \cdot 2.04 \approx 35.34 \, \text{m} \)

Particle 2: Speed 9 m/s at 60 Degrees

• Initial Velocity Components:
  • Horizontal Component: \( V_{2x} = V_2 \cdot \cos(\theta_2) = 9 \cdot \cos(60^\circ) = 9 \cdot \frac{1}{2} = 4.5 \, \text{m/s} \)
  • Vertical Component: \( V_{2y} = V_2 \cdot \sin(\theta_2) = 9 \cdot \sin(60^\circ) = 9 \cdot \frac{\sqrt{3}}{2} \approx 7.79 \, \text{m/s} \)
• Time of Flight: Similarly, we calculate the time of flight \( T_2 \):
  • \( T_2 = \frac{2 \cdot 7.79}{9.81} \approx 1.59 \, \text{s} \)
• Horizontal Range: The horizontal distance covered by Particle 2 is:
  • \( R_2 = V_{2x} \cdot T_2 = 4.5 \cdot 1.59 \approx 7.15 \, \text{m} \)

Calculating the Maximum Distance Between the Two Particles

To find the maximum distance between the two particles until they hit the ground, we need to consider their positions at the time of their respective landings. The horizontal distance between them at the time of landing is given by:

• Distance Between Particles:
  • \( D = R_1 - R_2 = 35.34 - 7.15 \approx 28.19 \, \text{m} \)

Final Thoughts

The maximum distance between the two particles until they strike the ground is approximately 28.19 meters. This analysis highlights how different speeds and angles of projection can significantly affect the trajectories and distances traveled by projectiles. Understanding these principles is crucial in physics, especially in the study of motion and forces.