Question

A roller coaster is moving at \(2.00 \mathrm{~m} / \mathrm{s}\) at the top of the first hill \((h=40.0 \mathrm{~m}) .\) Ignoring friction and air resistance, how fast will the roller coaster be moving at the top of a subsequent hill, which is \(15.0 \mathrm{~m}\) high?

Short answer

Answer: The final speed of the roller coaster at the top of the second hill is approximately 23.96 m/s.

Step-by-step solution

Calculate the initial kinetic and potential energy of the roller coaster at the top of the first hill.

First, we need to find the initial kinetic energy (KE) and potential energy (PE) of the roller coaster at the top of the first hill. The initial kinetic energy can be calculated using the formula KE = 0.5 * m * v^2, where m is the mass of the roller coaster and v is its speed. We are not given the mass of the roller coaster, but since the mass remains the same throughout the problem, we can work with the expression 0.5 * m * v^2. The initial potential energy can be calculated using the formula PE = m * g * h, where g is the acceleration due to gravity (9.81 m/s²) and h is the height of the first hill. So we have: Initial KE = 0.5 * m * (2.00 m/s)^2 Initial PE = m * (9.81 m/s²) * (40.0 m)

Determine the kinetic energy of the roller coaster at the top of the second hill using mechanical energy conservation.

According to the conservation of mechanical energy principle, the sum of the kinetic and potential energies of the roller coaster at the top of the first and second hills should be equal. So we can write: Initial KE + Initial PE = Final KE + Final PE At the top of the second hill, the height is 15.0 m, so we can calculate the final potential energy: Final PE = m * (9.81 m/s²) * (15.0 m) Now, we can substitute the expressions for initial and final potential energy into the conservation of mechanical energy equation: 0.5 * m * (2.00 m/s)^2 + m * (9.81 m/s²) * (40.0 m)= Final KE + m * (9.81 m/s²) * (15.0 m) We can solve this equation for the final kinetic energy: Final KE = 0.5 * m * (2.00 m/s)^2 + m * (9.81 m/s²) * (40.0 m) - m * (9.81 m/s²) * (15.0 m)

Calculate the final speed of the roller coaster at the top of the second hill.

Now that we have the expression for the final kinetic energy, we can calculate the final speed of the roller coaster. The final KE can be written as: Final KE = 0.5 * m * v_f^2 where v_f is the final speed we want to find. Solving this equation for v_f, we have: v_f^2 = 2 * (0.5 * m * (2.00 m/s)^2 + m * (9.81 m/s²) * (40.0 m) - m * (9.81 m/s²) * (15.0 m))/(m) We can see that m, the mass of the roller coaster, cancels out: v_f^2 = 2 * (0.5 * (2.00 m/s)^2 + (9.81 m/s²) * (40.0 m) - (9.81 m/s²) * (15.0 m)) Now, we can calculate the final speed: v_f = sqrt(2 * (0.5 * (2.00 m/s)^2 + (9.81 m/s²) * (40.0 m) - (9.81 m/s²) * (15.0 m))) v_f ≈ 23.96 m/s Thus, the roller coaster will be moving at approximately 23.96 m/s at the top of the second hill, which is 15.0 m high.

Key concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It depends on the mass and the velocity of the object and is given by the formula \( KE = \frac{1}{2} m v^2 \), where \( m \) is the mass and \( v \) is the velocity. In our roller coaster example, we have a starting speed of 2.00 m/s. Even though the mass \( m \) of the roller coaster isn't provided, the relationship between speed and kinetic energy allows us to focus on how changes in speed affect kinetic energy.

When considering kinetic energy in problem-solving, always account for speed changes because they have a significant impact, given that kinetic energy is proportional to the square of the velocity. This means even small changes in speed can result in large changes in kinetic energy. For our problem, this concept helps us anticipate that a higher speed at another location will result in a higher kinetic energy at that point.

Potential Energy

Potential energy is energy stored due to an object's position or height relative to a reference point, typically in a gravitational field. It can be calculated using the formula \( PE = mgh \), where \( m \) is mass, \( g \) is the acceleration due to gravity (approximately 9.81 m/s²), and \( h \) is the height.

In the context of the roller coaster, the initial potential energy is determined by the height of 40.0 m of the first hill. As the roller coaster moves to a subsequent hill of 15.0 m height, the potential energy will decrease due to the lower position.
Understanding potential energy helps explain how the roller coaster can speed up as it moves downhill. The decrease in height reduces potential energy and increases kinetic energy, showing the interchangeability of energy forms in a closed system where no external work is done.

Mechanical Energy Conservation

The conservation of mechanical energy principle states that the total mechanical energy (sum of kinetic and potential energy) in a closed system remains constant if there is no external force like friction doing work on it. Mathematically, we express this as \( KE_i + PE_i = KE_f + PE_f \), where \( i \) and \( f \) denote the initial and final states, respectively.

In our exercise, this principle helps us determine the final speed of the roller coaster at the top of the second hill. Initially, the roller coaster has specific kinetic and potential energies at the higher point (40.0 m). As it descends, potential energy decreases while kinetic energy increases, maintaining the sum (total energy) equal at all times. At the second hill, even without calculating mass, we can solve for the roller coaster's final speed knowing the exchange between potential and kinetic energy, ultimately confirming conservation of mechanical energy when friction is neglected.

• Ensure you identify initial and final energy states correctly.
• Note how energy transfers between forms as height changes.
• Understand that energy conservation aids in calculating unknowns, like speed or height, in dynamic systems.