A roller-coaster car is moving at
20.0 m/s along a straight horizontal track. What is its speed after climbing a 15.0 m hill?
10
To determine the roller-coaster car's speed after climbing a 15.0 m hill, you can use the principle of conservation of energy. The initial kinetic energy (KE) of the car will be converted into potential energy (PE) as it climbs the hill.
First, we can find the initial kinetic energy (KE₁) of the car using the formula:
KE₁ = (1/2) * mass * velocity²
Since we are not given the mass of the car, we can simply assume a value of 1 kg for convenience. Therefore, the initial kinetic energy will be:
KE₁ = (1/2) * 1 kg * (20.0 m/s)² = 200 J
As the car climbs the hill, this initial kinetic energy will be converted entirely into potential energy at the peak of the hill. Therefore, we can equate the initial kinetic energy to the potential energy at the peak of the hill:
PE = KE₁ = m * g * height
Where:
- PE is the potential energy at the peak of the hill
- m is the mass of the car
- g is the acceleration due to gravity (9.8 m/s²)
- height is the height of the hill (15.0 m)
Since we can assume a mass of 1 kg, we can rearrange the equation to find the potential energy:
PE = m * g * height = 1 kg * 9.8 m/s² * 15.0 m = 147 J
The potential energy, which is equal to the initial kinetic energy, can now be used to find the final velocity (v₂) of the car using the formula:
KE₂ = (1/2) * mass * velocity₂²
Since the potential energy is equal to the kinetic energy, we can set them equal to each other:
KE₂ = 147 J
(1/2) * 1 kg * velocity₂² = 147 J
velocity₂² = 2 * (147 J) / (1 kg)
velocity₂² = 294 m²/s²
Taking the square root of both sides:
velocity₂ = √(294 m²/s²)
velocity₂ ≈ 17.15 m/s
Therefore, the roller-coaster car's speed after climbing the 15.0 m hill is approximately 17.15 m/s.