Final answer:
The final speed of the rock just before it hits the water can be found using the conservation of energy, yielding approximately 24.3 m/s. This speed is independent of the direction of the throw due to energy considerations.
Step-by-step explanation:
Calculating the Speed of a Rock Thrown from a Bridge
To find the speed of the rock just before it hits the water, we will use the principle of conservation of energy. This physics principle states that the total mechanical energy (kinetic plus potential) in a system remains constant if only conservative forces (like gravity) are doing work.
Initially, when the rock is thrown from the bridge, it has both kinetic energy (due to its initial speed) and gravitational potential energy (due to its height above the water). As it falls, potential energy is converted into kinetic energy.
The initial kinetic energy (Ki) can be given as:
Ki = 0.5 * m * vi2
The potential energy (Ui) at the height h is:
Ui = m * g * h
Just before the rock hits the water, the potential energy is zero and all the energy is kinetic. The final kinetic energy (Kf) is therefore equal to the sum of the initial kinetic and potential energies:
Kf = Ki + Ui
0.5 * m * vf2 = 0.5 * m * vi2 + m * g * h
When we solve for vf, the mass (m) cancels out, and we get:
vf = sqrt(vi2 + 2 * g * h)
Substituting the given values, we have:
vf = sqrt(16.42 + 2 * 9.8 * 16.4)
vf = sqrt(268.96 + 321.44)
vf = sqrt(590.4)
vf = 24.3 m/s (approximately)
This calculation shows that the final speed of the rock just before it hits the water is approximately 24.3 m/s, and it is indeed independent of the direction the rock was thrown.