Final answer:
The 10 kg block will be moving at approximately 7.67 m/s at the bottom of a 3.0 m tall frictionless ramp, calculated by using the conservation of energy principle.
Step-by-step explanation:
To calculate how fast a 10 kg block will be moving at the bottom of a 3.0 m tall frictionless ramp using energy concepts, we begin by understanding that the potential energy at the top of the ramp will be converted into kinetic energy at the bottom. Since there's no friction, we can confidently use the conservation of mechanical energy principle. The potential energy (PE) can be calculated as PE = mgh, where m is the mass, g is acceleration due to gravity (9.81 m/s²), and h is the height of the ramp.
For the given block, PE = (10 kg) * (9.81 m/s²) * (3.0 m) = 294.3 J. At the bottom of the ramp, all this potential energy will have been converted into kinetic energy (KE), given by KE = 0.5 * m * v², where v is the velocity, we want to find.
By setting the PE equal to KE, we can solve for v: 294.3 J = 0.5 * (10 kg) * v², simplifying to v² = (294.3 J * 2) / 10 kg, giving v² = 58.86 m²/s². Hence, the velocity v is the square root of 58.86 m²/s², which is approximately 7.67 m/s. Therefore, the block will be moving at 7.67 m/s at the bottom of the ramp.