A partical at a height H from the ground is projected with an angle 30⁰ from the horizontal,it strikes the ground making angle 45⁰ with horizontal.It is again projected from the same point at height h with the same speed but with an angle of 60⁰ with the horizontal. Find the angle it make with the horizontal when it strikes the ground.

To solve this problem, we need to analyze the motion of the projectile in two different scenarios. The first scenario involves a particle projected from a height H at an angle of 30 degrees, and it strikes the ground at an angle of 45 degrees. In the second scenario, the same particle is projected from a height h at an angle of 60 degrees. Our goal is to find the angle it makes with the horizontal when it strikes the ground in this second case.

Understanding Projectile Motion

Projectile motion can be broken down into two components: horizontal and vertical. The horizontal motion is uniform, while the vertical motion is influenced by gravity. The angle of projection affects the trajectory and the time of flight.

First Scenario Analysis

In the first case, the particle is projected at an angle of 30 degrees. The initial velocity can be broken down into horizontal (Vx) and vertical (Vy) components:

• Vx = V * cos(30°)
• Vy = V * sin(30°)

When the particle strikes the ground at an angle of 45 degrees, we can use the relationship between the vertical and horizontal components of the velocity at the point of impact. At 45 degrees, the magnitudes of the horizontal and vertical components of the velocity are equal:

• Vx_final = Vy_final

Using the equations of motion and the known angle of impact, we can derive the relationship between the height H and the initial velocity V. However, we will focus on the second scenario for our main calculation.

Second Scenario Analysis

Now, let's analyze the second scenario where the particle is projected from height h at an angle of 60 degrees. The initial velocity components are:

• Vx = V * cos(60°)
• Vy = V * sin(60°)

At the point of impact, we need to determine the angle with the horizontal. The vertical component of the velocity just before impact can be calculated using the kinematic equations, considering the height h from which it is projected.

Calculating the Final Vertical Velocity

Using the kinematic equation for vertical motion:

• Vy_final^2 = Vy_initial^2 + 2 * g * h

Substituting the values:

• Vy_final^2 = (V * sin(60°))^2 + 2 * g * h

Now, we can find the final horizontal velocity, which remains constant throughout the motion:

• Vx_final = V * cos(60°)

Finding the Angle of Impact

The angle θ with the horizontal at the point of impact can be found using the tangent function:

• tan(θ) = Vy_final / Vx_final

Substituting the expressions for Vy_final and Vx_final, we can find the angle θ:

• tan(θ) = sqrt((V * sin(60°))^2 + 2 * g * h) / (V * cos(60°))

From this equation, we can calculate θ, which will give us the angle at which the particle strikes the ground after being projected from height h at an angle of 60 degrees.

Conclusion

By following these steps, we can derive the angle of impact for the second scenario. The key takeaway is that the angle of projection and the height from which the particle is launched significantly influence the trajectory and the angle at which it strikes the ground. This problem illustrates the principles of projectile motion and the importance of understanding both the vertical and horizontal components of motion.