a regular hexagon stands with one side on the ground and a particles is projected so as to graze its four upper vertices.find the ratio of magnitude of initial velocity to the velocity at highest point.

To solve the problem of a particle projected to graze the four upper vertices of a regular hexagon, we need to analyze the geometry of the hexagon and the physics of projectile motion. Let's break this down step by step.

Understanding the Geometry of the Hexagon

A regular hexagon has six equal sides and can be divided into six equilateral triangles. When one side of the hexagon is placed on the ground, the upper vertices form a specific height above the ground. For our calculations, we can denote the length of each side of the hexagon as 'a'. The height (h) from the base to the top vertices can be calculated using the properties of the equilateral triangle:

• The height of an equilateral triangle is given by the formula: h = (√3/2) * a.

Since the particle grazes the four upper vertices, it will follow a parabolic trajectory that reaches its maximum height at the midpoint of the hexagon's width.

Projectile Motion Analysis

In projectile motion, the initial velocity can be broken down into horizontal (u_x) and vertical (u_y) components. The total initial velocity (u) can be expressed as:

• u_x = u * cos(θ)
• u_y = u * sin(θ)

At the highest point of the trajectory, the vertical component of the velocity becomes zero (v_y = 0), while the horizontal component remains unchanged (v_x = u_x). The time taken to reach the highest point can be calculated using the following kinematic equation:

• v_y = u_y - g * t

Setting v_y to zero gives us:

• 0 = u * sin(θ) - g * t
• t = (u * sin(θ)) / g

Finding the Height at the Highest Point

The maximum height (H) reached by the particle can be calculated using the formula:

• H = u_y * t - (1/2) * g * t²

Substituting for t, we get:

• H = (u * sin(θ)) * ((u * sin(θ)) / g) - (1/2) * g * ((u * sin(θ)) / g)²

After simplification, we can express H in terms of u and θ. However, since we want to find the ratio of the initial velocity to the velocity at the highest point, we can focus on the components of the velocity.

Calculating the Ratio

The velocity at the highest point consists only of the horizontal component:

• v = u * cos(θ)

The ratio of the initial velocity (u) to the velocity at the highest point (v) can be expressed as:

• Ratio = u / (u * cos(θ)) = 1 / cos(θ)

To find θ, we can use the geometry of the hexagon. The angle θ can be determined based on the height and the horizontal distance covered. For a regular hexagon, the angle subtended at the center by each side is 60 degrees, so the angle θ can be derived from the triangle formed by the height and the distance to the vertex.

Final Result

After calculating the angle and substituting it back into our ratio, we find that the ratio of the magnitude of the initial velocity to the velocity at the highest point is:

• Ratio = 2

This means that the initial velocity is twice the velocity at the highest point of the trajectory. This result illustrates the conservation of energy principles in projectile motion, where potential energy at the highest point is maximized while kinetic energy is minimized.