Step 1: Find the speed of the ball w.r.to the ground in the two cases.
In this problem, we have to keep in mind the frame of the observer. Here, we must be clear that we are seeing the motion from the ground. Compare to the velocity of the train (\(10\) m/s), the speed of the ball is less (\(1\) m/s).
When the ball is moving in the direction of the train, the speed of the ball with respect to ground = \(10+1=11\) m/s in the direction of the train
When the ball is moving opposite to the direction of the train, the speed of the ball with respect to ground = \(10-1=9\) m/s in the direction of the train
So the direction of the ball does not change w.r.t. ground as the speed of the train is more than the speed of the ball.

Step 2: Find the time after which the speed of the ball changes.
As the speed is changing after travelling \(10\) m and the speed of the ball is \(1\) m/s w.r.t. the train, hence, the time duration of the changing speed = \(10\) sec
Since the collision of the ball is perfectly elastic there is no dissipation of energy, hence, total momentum and kinetic energy are conserved.

Step 3: Find the average speed of the ball after each \(20\) sec.
In the first \(10\) sec, the distance travelled by the ball = \(11\times 10=110~\text{m}\)
In the next \(10\) sec, the distance travelled by the ball = \(9\times 10= 90~\text{m}\)
The total distance travelled by the ball in each \(20\) sec will be the same and is equal to \((110+90) 200~\text{m}\).
So, the average speed of the ball will be the same after each interval of \(20\) sec.

Since the train is moving with constant velocity, hence, it will act as an inertial frame of reference as that of Earth and acceleration will be the same in both frames.
We should not confuse it with the non-inertial and inertial frames of reference. A frame of reference that is not accelerating will be inertial.
Therefore, the correct options are (b), (c), and (d).
Hence, option (3) is the correct answer.