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A ball falls from rest at a height H above a lake. Let y = 0 at the surface of the lake. As the ball falls, it experiences a gravitational force -mg. When it enters the water, it experiences a buoyant force B so the net force in the water is B - mg.

a) Write an expression for v(t) and y(t) while the ball is falling in air.

b) In the water, let v_{2}(t) = at + b and y_{2}(t) = (1/2)at^{2}+ bt + c where a = (B - mg)/m. Use

continuity conditions at the surface of the water to find the constants b and c.

The attempt at a solution

a)

Since [itex]\vec{v}=\int\vec{a}dt=\vec{a}t+\vec{v}_{0}[/itex] and the initial velocity is 0, we have [itex]\vec{v}(t)=\vec{a}t[/itex]. Using [itex]\vec{F}=m\vec{a}[/itex] yields [itex]v(t)=-gt[/itex].

Also, [itex]\vec{r}=\int\vec{v}dt=\frac{1}{2}\vec{a}t^{2}+\vec{v}t+\vec{r}_{0}[/itex]. Again, since the initial velocity is 0 and [itex]y=r-r_{0}[/itex] we have [itex]y(t)=-\frac{1}{2}gt^{2}[/itex].

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# Homework Help: Continuity Conditions

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