The speed is 9.2 m/s
Explanation:
According to the law of conservation of energy, the total mechanical energy of the rock during the fall is conserved:
[tex]E=U+K[/tex]
where
[tex]U=mgh[/tex] is the potential energy, with
m being the mass of the rock
g is the acceleration of gravity
h is the height of the rock relative to the ground
[tex]K=\frac{1}{2}mv^2[/tex] is the kinetic energy, with
v being the speed of the rock
When the kinetic energy is equal to the gravitational potential energy,
[tex]mgh' = \frac{1}{2}mv^2[/tex]
which means
[tex]v=\sqrt{2gh'}[/tex] (1)
Since the potential energy is proportional to the height h, this situation will occur when the potential energy has halved, therefore when the height has halved compared to the initial height:
[tex]h'=\frac{h}{2}=\frac{8.7}{2}=4.35 m[/tex]
Therefore, the speed at which this occurs is
[tex]v=\sqrt{2(9.8)(4.35)}=9.2 m/s[/tex]
Learn more about potential energy and kinetic energy:
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