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glebovg
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Homework Statement
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 v2(t) = at + b and y2(t) = (1/2)at2 + 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 \vec{v}=\int\vec{a}dt=\vec{a}t+\vec{v}_{0} and the initial velocity is 0, we have \vec{v}(t)=\vec{a}t. Using \vec{F}=m\vec{a} yields v(t)=-gt.
Also, \vec{r}=\int\vec{v}dt=\frac{1}{2}\vec{a}t^{2}+\vec{v}t+\vec{r}_{0}. Again, since the initial velocity is 0 and y=r-r_{0} we have y(t)=-\frac{1}{2}gt^{2}.
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 v2(t) = at + b and y2(t) = (1/2)at2 + 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 \vec{v}=\int\vec{a}dt=\vec{a}t+\vec{v}_{0} and the initial velocity is 0, we have \vec{v}(t)=\vec{a}t. Using \vec{F}=m\vec{a} yields v(t)=-gt.
Also, \vec{r}=\int\vec{v}dt=\frac{1}{2}\vec{a}t^{2}+\vec{v}t+\vec{r}_{0}. Again, since the initial velocity is 0 and y=r-r_{0} we have y(t)=-\frac{1}{2}gt^{2}.
