Problem Statement:
Consider a uniform spherical planet of mass M and radius R. Two parallel tunnels are dug perpendicular distance symmetrically from centre. A particle is to be thrown from the starting of one tunnel such that it enters in another tunnel without making collision with tunnel wall and continues its motion. Tunnels are frictionless and particle just fits inside the tunnel. Consider the planet to be at rest.

Solution:

To solve this problem, we need to consider the following steps:

Step 1: Determine the gravitational field inside the planet at a distance d from the center.

Step 2: Determine the speed at which the particle should be thrown from the starting point of the tunnel.

Step 3: Determine the trajectory of the particle as it travels through the tunnel and exits into the other tunnel.

Step 1: Gravitational Field inside the Planet

The gravitational field inside a uniform spherical planet can be determined using the equation:

g = (GM/r^3) * r

where g is the gravitational field, M is the mass of the planet, r is the distance from the center of the planet, and G is the gravitational constant.

At a distance d from the center of the planet, the gravitational field can be determined as:

g = (GM/d^3) * d

Step 2: Speed of the Particle

The speed at which the particle should be thrown from the starting point of the tunnel can be determined using the equation for circular motion:

v = sqrt(g * d)

where v is the velocity of the particle, g is the gravitational field at a distance d from the center of the planet, and d is the distance between the two tunnels.

Step 3: Trajectory of the Particle

The trajectory of the particle can be determined by considering the motion of the particle as it travels through the tunnel and exits into the other tunnel. Since the tunnels are frictionless and the particle just fits inside the tunnel, we can assume that the particle will move in a straight line.

As the particle travels through the first tunnel, it will experience a gravitational force towards the center of the planet. This force will cause the particle to move in a curved path towards the center of the planet. However, since the particle is moving with a sufficient velocity, it will not collide with the wall of the tunnel.

As the particle exits the first tunnel and enters the second tunnel, it will continue to move in a straight line. However, since the second tunnel is also curved towards the center of the planet, the particle will experience a gravitational force towards the center of the planet. This force will cause the particle to move in a curved path towards the center of the planet. Again, since the particle is moving with a sufficient velocity, it will not collide with the wall of the tunnel.

Finally, as the particle exits the second tunnel, it will continue to move in a straight line and travel away from the planet.