Rock Boat Solution

The answer is that there are three cases to consider: the rock can be denser than water (case 1), the same density as water (case 2) or less dense than water (case 3). Each of these cases needs to be looked at. Let us look at what the situation looks like before we throw the rock into the water.

The amount of water needed to be displaced (so the boat and rock can both float) before the rock is tossed is given by Vd. The density of the water is given by pw, g stands for gravity. Therefore, for the boat to float witht the rock in it
pw*Vd*g=(Mb+Mr)*g
Where the two mass terms stand for the mass of the boat (Mb) and the mass of the rock (Mr). So the total amount of water displaced initially (Vd) is given by
Vd=(Mb+Mr)/pw
After the rock is thrown over board, the amount of water the boat displaces will equal V=Mb/pw
The amount of water displaced by the rock will now equal Vr (this is only true for cases 1 and 2), which from the definition of density ( p = M/V) Vr=Mr/pr
So the total amount of water displaced after the rock is thrown (Va) is
Va = Mb/pw + Mr/pr
The difference between the final and initial volumes is diff in V = (Mb*pr+Mr*pw)/(pw*pr)-(Mb+Mr)/pw=Mr*(pw-pr)/(pw*pr)
Case 1: rock denser than water the change in volume is negative (initial volume required to hold up the boat was larger than the final volume so the level of the water decreases)
Case 2: rock equal to water in density so the change in volume is zero (no chnage in the level of the water)
For case three we cannot use the formulism used for the first two cases as we were assuming the rock would be sinking under the sureface of the water.
We know after we throw the rock in the volume displaced by the boat will be V=Mb/pw and the volume of water displaced by the rock in order to keep it afloat (remember we know it is floating becasue it is less dense than water) will be V=Mr/pw
Summing the total volume of water required to keep the rock and boat afloat
Va = (Mb+Mr)/pw which is equal to the volume of water required to keep the boat and rock afloat before we threw the rock out --> for case 3 the level of the water does not change.

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