1. **Problem Statement:** Design a rollercoaster with the following parameters: - Height of the tallest point: 198 m - Maximum speed: 251 km/h - Initial height: 90 feet (convert to meters) - Calculate energy and forces for acceleration and reaching the highest point. 2. **Given Data and Conversions:** - Mass of average person: 137 pounds $\times 8$ people - Mass of rollercoaster car: 1300 kg - Total mass of train: 493,512 kg - Initial height: 90 feet $= 90 \times 0.3048 = 27.432$ m - Maximum speed: 251 km/h $= \frac{251 \times 1000}{3600} = 69.72$ m/s 3. **Formulas and Important Rules:** - Gravitational potential energy: $$PE = mgh$$ where $m$ is mass, $g = 9.8\, m/s^2$, $h$ is height. - Kinetic energy: $$KE = \frac{1}{2}mv^2$$ where $v$ is velocity. - Total mechanical energy at highest point equals energy at end of acceleration. 4. **Step 1: Calculate mass of average person in kg** $$137 \text{ pounds} = 137 \times 0.4536 = 62.14\, kg$$ Total mass of people: $$62.14 \times 8 = 497.12\, kg$$ 5. **Step 2: Calculate total mass of train** Given total mass is 493,512 kg (likely including cars and people). 6. **Step 3: Calculate energy needed to reach highest point (198 m):** $$PE = mgh = 493,512 \times 9.8 \times 198 = 9.57 \times 10^8\, J$$ 7. **Step 4: Calculate speed needed at end of acceleration to reach 198 m:** Energy at end of acceleration is kinetic energy: $$KE = PE \Rightarrow \frac{1}{2}mv^2 = mgh$$ Cancel mass $m$: $$\frac{1}{2}v^2 = gh$$ $$v^2 = 2gh$$ $$v = \sqrt{2 \times 9.8 \times 198} = \sqrt{3880.8} = 62.3\, m/s$$ 8. **Step 5: Calculate force needed to accelerate train to 62.3 m/s:** Assuming acceleration over distance $d$ (length of acceleration track), force $F$ is related to work done: $$W = Fd = \Delta KE = \frac{1}{2}mv^2$$ Rearranged: $$F = \frac{\frac{1}{2}mv^2}{d}$$ Since $d$ is not given, force depends on chosen acceleration length. **Summary:** - Mass of average person: 62.14 kg - Total mass of train: 493,512 kg - Initial height: 27.432 m - Energy to reach 198 m: $$9.57 \times 10^8\, J$$ - Speed needed at end of acceleration: 62.3 m/s - Force needed depends on acceleration distance. This completes the calculations for the first problem.