1. Problem 7: Find the velocity of an object dropped from rest falling 25 meters, neglecting air resistance. 2. Use the kinematic equation for free fall velocity: $$v = \sqrt{2gh}$$ where $g = 9.8$ m/s$^2$ (acceleration due to gravity) and $h = 25$ m (height fallen). 3. Calculate velocity: $$v = \sqrt{2 \times 9.8 \times 25} = \sqrt{490} \approx 22.14 \text{ m/s}$$ 4. Problem 8: Find the maximum speed of a 2.0 kg ball launched by a spring compressed 0.25 m with spring constant $k=1000$ N/m. 5. Use conservation of energy: potential energy stored in spring converts to kinetic energy of ball. 6. Spring potential energy: $$PE = \frac{1}{2}kx^2 = \frac{1}{2} \times 1000 \times (0.25)^2 = 31.25 \text{ J}$$ 7. Kinetic energy of ball: $$KE = \frac{1}{2}mv^2$$ 8. Set $PE = KE$ and solve for $v$: $$31.25 = \frac{1}{2} \times 2.0 \times v^2 \Rightarrow v^2 = \frac{31.25 \times 2}{2.0} = 31.25 \Rightarrow v = \sqrt{31.25} \approx 5.59 \text{ m/s}$$ 9. Problem 9a: Write energy conservation statement for a spring launching a ball vertically upward in a frictionless closed system. 10. Energy conservation statement: $$\text{Initial total energy} = \text{Final total energy}$$ $$\frac{1}{2}kx^2 = mg(h)$$ where $k$ is spring constant, $x$ is compression, $m$ is mass, $g$ is gravity, and $h$ is height risen. 11. Problem 9b: Find how high a 0.5 kg ball rises when launched by a spring with $k=1000$ N/m compressed 0.25 m. 12. Calculate spring potential energy: $$PE = \frac{1}{2}kx^2 = \frac{1}{2} \times 1000 \times (0.25)^2 = 31.25 \text{ J}$$ 13. Set $PE = mgh$ and solve for $h$: $$31.25 = 0.5 \times 9.8 \times h \Rightarrow h = \frac{31.25}{0.5 \times 9.8} = \frac{31.25}{4.9} \approx 6.38 \text{ m}$$ 14. Problem 10: List two problem-solving approaches for a brick falling from 20 m chimney to find speed on impact. 15. Approaches: - Use kinematic equations: $$v = \sqrt{2gh}$$ - Use energy conservation: potential energy converts to kinetic energy $$mgh = \frac{1}{2}mv^2$$ Final answers: 7. Velocity = 22.14 m/s 8. Max speed = 5.59 m/s 9b. Height risen = 6.38 m 10. Approaches: kinematics and energy conservation