A 1000 kg roller coaster begins on a 10 m tall hill with an initial velocity of 6m/s and travels down before traveling up a second hill. As the coaster moves from its initial height to its lowest position, 1700J of energy is transformed to thermal energy by friction. In order for the roller coaster to safely travel over the second hill, it must be moving at a velocity of 4.6m/s or less at the top of the second hill. What is the maximum height the second hill can be?
To solve this problem, we can use the law of conservation of energy. The roller coaster starts with potential energy at the top of the first hill and converts it to kinetic energy as it moves down the hill. Then, some of that kinetic energy is converted to thermal energy due to friction. Finally, the remaining kinetic energy allows the roller coaster to go up the second hill.
1. Calculate the initial potential energy at the top of the first hill:
Potential energy (PE) = mass (m) × acceleration due to gravity (g) × height (h1)
Given: m = 1000 kg, g = 9.8 m/s^2, h1 = 10 m
PE1 = 1000 kg × 9.8 m/s^2 × 10 m
PE1 = 98,000 J
2. Calculate the initial kinetic energy at the bottom of the first hill:
Kinetic energy (KE) = 0.5 × mass (m) × velocity^2 (v)
Given: m = 1000 kg, v = 6 m/s
KE1 = 0.5 × 1000 kg × (6 m/s)^2
KE1 = 18,000 J
3. Calculate the energy lost due to friction:
Energy lost = PE1 - KE1
Energy lost = 98,000 J - 18,000 J
Energy lost = 80,000 J
4. Calculate the remaining kinetic energy at the top of the second hill (which needs to be 4.6 m/s or less):
KE2 ≤ 0.5 × mass (m) × velocity^2 (v)
Given: m = 1000 kg, v = 4.6 m/s
KE2 ≤ 0.5 × 1000 kg × (4.6 m/s)^2
KE2 ≤ 21,080 J
5. Calculate the maximum potential energy at the top of the second hill:
PE2 = KE1 + energy lost - KE2
PE2 = 18,000 J + 80,000 J - 21,080 J
PE2 = 76,920 J
6. Calculate the maximum height (h2) of the second hill using the maximum potential energy:
PE2 = m × g × h2
76,920 J = 1000 kg × 9.8 m/s^2 × h2
h2 = 76,920 J / (1000 kg × 9.8 m/s^2)
h2 ≈ 7.8 m
Therefore, the maximum height of the second hill should be approximately 7.8 meters.
To calculate the maximum height of the second hill, we need to consider the conservation of mechanical energy.
The initial total mechanical energy of the roller coaster at the top of the first hill is given by the sum of its gravitational potential energy and its kinetic energy:
E_initial = m * g * h + 0.5 * m * v_initial^2
where m is the mass of the roller coaster (1000 kg), g is the acceleration due to gravity (9.8 m/s^2), h is the height of the first hill (10 m), and v_initial is the initial velocity of the roller coaster (6 m/s).
The final total mechanical energy of the roller coaster at the top of the second hill is given by the sum of its gravitational potential energy and its kinetic energy:
E_final = m * g * h_final + 0.5 * m * v_final^2
where h_final is the height of the second hill and v_final is the velocity of the roller coaster at the top of the second hill (4.6 m/s).
Since there is friction between the roller coaster and the track that transforms 1700 J of energy into thermal energy, the change in total mechanical energy can be expressed as:
ΔE = E_initial - E_final = -1700 J
Substituting the expressions for E_initial and E_final, we get:
m * g * h + 0.5 * m * v_initial^2 - (m * g * h_final + 0.5 * m * v_final^2) = -1700 J
Simplifying the equation, we can cancel out the mass m:
g * h + 0.5 * v_initial^2 - (g * h_final + 0.5 * v_final^2) = -1700 J
Rearranging the equation and isolating h_final, we have:
h_final = (g * h + 0.5 * v_initial^2 - 0.5 * v_final^2 + 1700) / g
Now we can substitute the values into the equation:
h_final = (9.8 * 10 + 0.5 * 6^2 - 0.5 * 4.6^2 + 1700) / 9.8
Calculating this expression, we find:
h_final ≈ 36.43 m
Therefore, the maximum height the second hill can be is approximately 36.43 meters.