if the earth is drilled through the centre of the core and if we drop a metal ball from one end what happens to the ball ?

In first half i.e going from surface to the centre it would continually accelerate as the gravitational force is in towards the centre but as soon as it crosses the centre it would start deccelerate as now the gravitational force would be in opposite direction to motion and hence it would just reach the other end ..

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If we drop a metal ball from one end the accelerates  some distance but at the centre of the earth it will stop it does not go to the other end because at the center of the earth the acceleration is zero thats there is no acceleration at the centre.

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the acceleration vanishes but the ball has a good velocity and it moves further to negate that velocity