Final answer:
The final speed of the 100-g toy car moving up a frictionless slope can be found by using the conservation of mechanical energy, equating the initial and final mechanical energy, and solving for the final velocity to show that it is 0.687 m/s.
Step-by-step explanation:
To determine the final speed of a 100-g toy car that moves up a frictionless slope and gains 0.180 m in altitude, we apply the law of conservation of mechanical energy. The initial mechanical energy of the system is the sum of the car's kinetic energy and potential energy. Since the slope is frictionless, no mechanical energy is lost, and thus, the final mechanical energy must be equal to the initial mechanical energy.
The initial kinetic energy (KE) when the car's initial speed is 2.00 m/s can be calculated using the equation KE = 0.5 × m × v^2, where m is the mass of the car (0.1 kg) and v is the velocity (2.00 m/s). The potential energy (PE) gained when the car rises to 0.180 m can be calculated using PE = m × g × h, where g is the acceleration due to gravity (9.8 m/s^2) and h is the height (0.180 m).
By equating the initial mechanical energy to the final mechanical energy and solving for the final velocity, we can show that the final speed of the toy car is 0.687 m/s.