1. **Problem:** An engineer is testing a new free-fall ride with mass $m=500$ kg dropping from height $h=40$ m. 2. **Energy types halfway down and at the bottom:** - At the top, the ride has maximum potential energy and zero kinetic energy. - Halfway down (at height $h/2=20$ m), the ride has both potential and kinetic energy. - At the bottom (height $0$), the ride has maximum kinetic energy and zero potential energy. 3. **Formulas:** - Potential energy: $$PE = mgh$$ where $g=9.8$ m/s$^2$ is gravity. - Kinetic energy: $$KE = \frac{1}{2}mv^2$$ 4. **Calculate energies halfway down:** - Potential energy halfway: $$PE = 500 \times 9.8 \times 20 = 98000 \text{ J}$$ - Total energy at top: $$E = 500 \times 9.8 \times 40 = 196000 \text{ J}$$ - Kinetic energy halfway: $$KE = E - PE = 196000 - 98000 = 98000 \text{ J}$$ 5. **Calculate speed halfway down:** - Using $$KE = \frac{1}{2}mv^2$$, solve for $v$: $$98000 = \frac{1}{2} \times 500 \times v^2$$ $$98000 = 250 v^2$$ $$v^2 = \frac{98000}{250} = 392$$ $$v = \sqrt{392} \approx 19.8 \text{ m/s}$$ 6. **At the bottom:** - Potential energy: $$PE = 0$$ - Kinetic energy equals total energy: $$KE = 196000 \text{ J}$$ - Speed at bottom: $$196000 = \frac{1}{2} \times 500 \times v^2$$ $$196000 = 250 v^2$$ $$v^2 = \frac{196000}{250} = 784$$ $$v = \sqrt{784} = 28 \text{ m/s}$$ 7. **How mass and speed affect kinetic energy:** - Kinetic energy is directly proportional to mass and to the square of speed. - Doubling mass doubles kinetic energy. - Doubling speed quadruples kinetic energy. 8. **How mass and height affect potential energy:** - Potential energy is directly proportional to mass and height. - Doubling mass or height doubles potential energy. **Final answers:** - Halfway down: ride has both potential and kinetic energy, each about 98000 J. - Bottom: ride has maximum kinetic energy 196000 J and zero potential energy. - Kinetic energy depends on mass and square of speed. - Potential energy depends on mass and height.