Final Answer:
The block will reach a height of 0.852 meters relative to where it started. This calculation involves the conservation of momentum during the dart-block collision and the subsequent conversion of kinetic energy into gravitational potential energy. Understanding these principles allows for a precise determination of the maximum height attained by the block-dart system.
Step-by-step explanation:
When the foam dart collides with the hanging block, the principle of conservation of momentum comes into play. Initially, the dart possesses momentum determined by its mass (0.0368 kg) multiplied by its initial speed (16.2 m/s). In the collision, this momentum is conserved, and the block-dart system moves together with a combined mass equivalent to the sum of the individual masses of the block and dart. The conservation of momentum allows us to determine the final velocity of the system after the collision.
Subsequently, the principle of conservation of energy is invoked to understand the height the block-dart system reaches. The potential energy gained by the system during the collision is converted into gravitational potential energy as the system ascends. Using the gravitational potential energy equation (mgh), where "m" is the system's mass, "g" is the acceleration due to gravity, and "h" is the height, we can derive the maximum height attained by the system.
In summary, the collision transfers momentum from the dart to the block, and the subsequent rise of the block-dart system is governed by the conversion of kinetic energy to potential energy. This interplay of momentum and energy conservation elucidates the behavior of objects in motion and their transformations in the realm of physics.