Range of cannon

Question:

The range of a shell fired from a cannon is !!1km!!. Then the range of second identical shell which breaks into two equal parts is? One of the parts recoils with the same speed before breaking.

Solution

In this question we are given the initial case where the shell does not break mid air. While in the second case the shell splits in mid air.

Since there is no external force the momemtum is going to be conserved.

So if we assume the velocity of the ball before splitting is !!v!!. Then after splitting one half is !!-v!! and other half is !!x!!.

We will get !!mv = -m/2 v+m/2x !!. Or !!x =3v!!.

Had it been going with a speed !!v!! the distance covered by shell from the highest point is !!1/2km!!.

Now if the speed is !!3v!! the distance covered by shell will be !!3/2km!! from the highest point. So the range is !!2km!!

PS: Be careful, here the vertical component of the velocity is zero so in both cases the time of flight is same.

Second approach:

A more mechanical approach would be to conserve the position of center of mass.

So center of mass falls at 1km from the cannon.

Since one part of shell is reversing its direction and will reach back the point from which it was launched.

So we !!m*1=m/2*0+m/2*L!!

!!L=2Km!!

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