Final answer:
To determine the height a roller coaster can climb with no friction and an initial speed of 24.0 m/s, the principle of conservation of energy is used. The roller coaster's kinetic energy is converted into potential energy, resulting in a calculable height of approximately 29.36 meters.
Step-by-step explanation:
The student is asking about the concept of conservation of energy in the context of a roller coaster car climbing a hill after moving at a certain speed. To find out the height the roller coaster can climb without the presence of friction, we can use the conservation of mechanical energy principle, which states that the kinetic energy (KE) at the bottom of the hill will be converted into potential energy (PE) at the top of the hill.
The initial kinetic energy (KE) of the car can be calculated using the formula KE = ½ mv², where 'm' is the mass of the car (850 kg) and 'v' is the velocity (24.0 m/s). The potential energy (PE) at the height 'h' the car can climb is given by PE = mgh, where 'g' is the acceleration due to gravity (9.81 m/s²). By setting KE equal to PE, we can solve for the height 'h'.
Therefore, the calculation is as follows:
- KE = ½ * 850 kg * (24.0 m/s)²
- PE = 850 kg * 9.81 m/s² * h
- KE = PE
- ½ * 850 kg * (24.0 m/s)² = 850 kg * 9.81 m/s² * h
- h = (½ * (24.0 m/s)² ) / 9.81 m/s²
- h = (0.5 * 576) / 9.81
- h = 288 / 9.81
- h ≈ 29.36 meters
So, the roller coaster should be able to climb up to approximately 29.36 meters high before coming to a stop if there is no friction.