a particle is projected with a velocity 'v' from an edge of a right circular cone with its radius 'r' with its vertical angle=90, placed on inclined plane

When a particle is projected from the edge of a right circular cone onto an inclined plane, we can analyze the motion using principles from physics, particularly kinematics and dynamics. Let’s break down the scenario step by step to understand the motion of the particle.

Understanding the Setup

Imagine a right circular cone with a vertical angle of 90 degrees, which means it has an open top and a circular base. The cone is placed on an inclined plane, and at the edge of the cone, a particle is projected with an initial velocity 'v'. The radius of the cone's base is 'r'.

Key Parameters

• Initial Velocity (v): This is the speed at which the particle is projected.
• Radius (r): The distance from the center of the cone's base to its edge.
• Incline Angle (θ): The angle of the inclined plane with respect to the horizontal.

Analyzing the Motion

To analyze the motion of the particle, we need to consider the forces acting on it and the trajectory it will follow after being projected. The key aspects to consider are:

1. Components of Velocity

When the particle is projected, its velocity can be broken down into two components: horizontal and vertical. If we assume the projection is at an angle θ with respect to the horizontal, we can express these components as:

• Horizontal Component (v_x): v_x = v * cos(θ)
• Vertical Component (v_y): v_y = v * sin(θ)

2. Motion on the Inclined Plane

As the particle travels through the air and eventually lands on the inclined plane, it will be subject to gravitational forces. The gravitational force can be resolved into two components relative to the incline:

• Perpendicular to the incline: F_perpendicular = mg * cos(θ)
• Parallel to the incline: F_parallel = mg * sin(θ)

3. Trajectory of the Particle

The trajectory of the particle can be modeled using the equations of motion. The time of flight until it hits the inclined plane can be calculated by considering the vertical motion under gravity. The equation of motion in the vertical direction is:

y = v_y * t - (1/2) * g * t²

where y is the vertical displacement, g is the acceleration due to gravity, and t is the time of flight.

Calculating the Impact Point

To find where the particle lands on the inclined plane, we need to set up the equations for both the horizontal and vertical motions. The horizontal distance traveled can be expressed as:

x = v_x * t

By substituting the expressions for v_x and v_y, we can solve for the time of flight and subsequently the impact point on the inclined plane.

Example Calculation

Let’s say the inclined plane makes an angle of 30 degrees with the horizontal. If the particle is projected with a velocity of 10 m/s, we can calculate:

• v_x = 10 * cos(30°) = 10 * (√3/2) ≈ 8.66 m/s
• v_y = 10 * sin(30°) = 10 * (1/2) = 5 m/s

Using these components, we can determine the time of flight and the horizontal distance traveled before the particle hits the inclined plane.

Final Thoughts

This scenario illustrates the principles of projectile motion and the effects of gravity on a particle projected from a cone onto an inclined surface. By breaking down the motion into components and applying the equations of motion, we can predict the trajectory and impact point of the particle effectively.