There is a square carom board of side`a'. A striker is projected in hole after two successive collisions. Assuming the collisions to be perfectly elastic and the surface to be smooth. The angle of projection of striker is:.....??the striker starts from mid-point of one side (the original question goes like this).

Can somebody draw a figure and help me solve the question.

Give a hint.

To tackle this problem, let's visualize the scenario first. Imagine a square carom board with a side length of 'a'. The striker starts from the midpoint of one side of the board and is projected towards the opposite side. Since the collisions are perfectly elastic and the surface is smooth, we can apply the principles of reflection and angles to determine the angle of projection after the two collisions.

Visualizing the Carom Board

Picture the square board divided into four equal sides. If we label the sides as A, B, C, and D in clockwise order, the striker begins at the midpoint of side A. When it is projected towards side C, it will collide with side B first, then with side C.

Understanding the Path of the Striker

When the striker hits side B, it will reflect off at the same angle it approached. This is a key property of elastic collisions. The angle of incidence equals the angle of reflection. After this first collision, the striker will then head towards side C, where it will collide again.

Breaking Down the Angles

Let’s denote the angle of projection from the midpoint of side A towards side B as θ. When the striker hits side B, it reflects off at angle θ. The path can be visualized as follows:

• The striker travels from the midpoint of side A to side B.
• Upon hitting side B, it reflects and travels towards side C.
• After hitting side C, it will either go into the hole or reflect again, depending on the angle.

Calculating the Angle of Projection

To find the angle of projection, we can use the properties of triangles and the fact that the angles in a triangle sum up to 180 degrees. When the striker hits side B, the angle of incidence (θ) will equal the angle of reflection. The key here is to realize that the total angle around the point of collision must also be considered.

Using Geometry

Since the striker starts from the midpoint of side A, the distance to side B is a/2. After reflecting off side B, the striker will travel towards side C. The angle formed with the horizontal (side A) can be calculated using trigonometric ratios or geometric properties of the square.

Hint for the Solution

To find the angle of projection after two collisions, consider the symmetry of the square and the equal angles formed during the reflections. The angle of projection can be derived from the relationship between the sides and the angles formed at each collision. You might want to set up a coordinate system to simplify your calculations.

By carefully analyzing the angles and using the properties of elastic collisions, you can derive the angle of projection of the striker after the two collisions. Remember, the key lies in understanding how the angles relate to each other through the reflections.