Final answer:
Using energy conservation, we can show that a rock thrown from a bridge with initial kinetic and potential energy will convert its potential energy to kinetic energy as it falls, resulting in a final speed that is independent of the initial throw direction.
Step-by-step explanation:
To show that a rock thrown from a bridge 20.0 m above water with an initial speed of 15.0 m/s strikes the water with a speed of 24.8 m/s irrespective of the direction thrown, we'll use the principles of conservation of energy, assuming negligible air resistance.
At the initial point when the rock is thrown, it has both kinetic and potential energy. The kinetic energy is due to its initial speed and the potential energy is due to its elevation above the water. As the rock falls, potential energy is converted into kinetic energy.
The total mechanical energy at the beginning (Ei) and the end (Ef) should be equal, given by:
Ei = Ki + Ui
Ef = Kf + Uf
Where:
- Ki is the initial kinetic energy
- Ui is the initial potential energy
- Kf is the final kinetic energy
- Uf is the final potential energy (which will be zero at the water's surface)
We know that:
- Ki = (1/2)mv^2
- Ui = mgh
- Kf = (1/2)mv^2 (to be found)
Substituting and solving for the final velocity, we find that the velocity when it hits the water is indeed 24.8 m/s, confirming that the original direction of the throw does not affect the final speed due to the conservation of mechanical energy.