Question

a 1200 kg frictionless roller coaster starts from rest at a height of 24 m. what is its kinetic energy when it goes over a hill that is 12 m high

Answer 1

Answer 2

Calculation:

1. Initial Potential Energy (PE₁):
   At the starting height (24 m), the roller coaster has only potential energy.

   $$P{E}_1=m\cdot g\cdot h_1=1200\text{kg}\times 9.8\text{m/s²}\times 24\text{m}=282,240\text{J}$$

2. Potential Energy at 12 m Hill (PE₂):
   When it reaches the 12 m hill, some potential energy remains.

   $$P{E}_2=m\cdot g\cdot h_2=1200\text{kg}\times 9.8\text{m/s²}\times 12\text{m}=141,120\text{J}$$

3. Kinetic Energy (KE) Calculation:
   Since energy is conserved (no friction), the lost potential energy converts to kinetic energy.

   $$KE=P{E}_1-P{E}_2=282,240\text{J}-141,120\text{J}=141,120\text{J}$$

Final Answer: The kinetic energy is 141,120 J

Answer 3

Absolutely! Let's solve this problem step-by-step using the principles of conservation of energy.

Given:

• Mass of the roller coaster (m) = 1200 kg

• Initial height (h₁) = 24 m

• Final height (h₂) = 12 m

• Initial velocity (v₁) = 0 m/s (starts from rest)

To find:

• Kinetic energy (KE₂) at the height of 12 m

Solution:

1. Calculate the initial potential energy (PE₁) at the starting height (24 m):

   PE₁ = m * g * h₁

   where:

   PE₁ = 1200 kg * 9.8 m/s² * 24 m

   PE₁ = 282,240 Joules

• m = mass (1200 kg)

• g = acceleration due to gravity (approximately 9.8 m/s²)

• h₁ = initial height (24 m)

2. Calculate the potential energy (PE₂) at the height of 12 m:

   PE₂ = m * g * h₂

   where:

   PE₂ = 1200 kg * 9.8 m/s² * 12 m

   PE₂ = 141,120 Joules

• m = mass (1200 kg)

• g = acceleration due to gravity (9.8 m/s²)

• h₂ = final height (12 m)

3. Apply the conservation of energy principle:

   In a frictionless system, the total mechanical energy (potential energy + kinetic energy) remains constant.

   Total Energy (initial) = Total Energy (final)

   PE₁ + KE₁ = PE₂ + KE₂

   Since the roller coaster starts from rest, KE₁ = 0.

   Therefore:

   PE₁ = PE₂ + KE₂

4. Solve for the kinetic energy (KE₂) at the height of 12 m:

   KE₂ = PE₁ - PE₂

   KE₂ = 282,240 Joules - 141,120 Joules

   KE₂ = 141,120 Joules

Answer:

The kinetic energy of the roller coaster when it goes over the hill that is 12 m high is 141,120 Joules.