A particle is projected with a certain velocity at an angle 'A' above d horizontal frm d foot of a given plane inclined at an angle of 45 to d horizontal. If d particle strikes d plane normally then tanA equals- A) 1/3 B) 1/2 C) 2^-1/2 D) 3

To solve the problem of a particle projected at an angle 'A' above the horizontal that strikes an inclined plane at an angle of 45 degrees, we need to analyze the relationship between the angles involved. The key here is to understand how the angle of projection and the angle of inclination interact when the particle strikes the plane normally.

Understanding the Geometry of the Situation

When a particle is projected, it follows a parabolic trajectory. The inclined plane, which is at a 45-degree angle to the horizontal, creates a specific geometric relationship with the trajectory of the particle. For the particle to strike the plane normally, the angle of incidence must equal the angle of reflection, which in this case means that the trajectory of the particle must be perpendicular to the plane at the point of impact.

Analyzing the Angles

Let’s denote the angle of projection as 'A' and the angle of inclination of the plane as 45 degrees. When the particle strikes the plane normally, the tangent of the angle of projection can be related to the angle of inclination. The relationship can be expressed using the tangent function:

• The angle of inclination of the plane is 45 degrees, so:
• tan(45 degrees) = 1

For the particle to strike the plane normally, the tangent of the angle of projection 'A' must equal the tangent of the angle of inclination. Therefore, we can set up the equation:

tan(A) = tan(45 degrees)

Setting Up the Equation

Since we know that tan(45 degrees) = 1, we can write:

tan(A) = 1

This implies that:

A = 45 degrees

Finding the Value of tan(A)

Now, we need to find the value of tan(A) in terms of the options provided. The options given are:

• A) 1/3
• B) 1/2
• C) 2^-1/2
• D) 3

Since we established that tan(A) = 1, we can compare this with the options. The only option that matches this value is:

None of the options provided directly equal 1. However, if we consider the context of the problem, we can conclude that the correct answer is not listed among the options. The angle A must be such that tan(A) = 1, which corresponds to A being 45 degrees.

Conclusion

In summary, for a particle projected at an angle 'A' to strike a plane inclined at 45 degrees normally, the tangent of angle A must equal 1. Therefore, the correct relationship is that tan(A) = 1, indicating that A = 45 degrees. If we were to choose from the given options, none would be correct based on this analysis.