✅ Adventure 5 — Solutions

1) How far, how fast?

• Distance at 5 s: x(5)=½·8·5² = 100 m
• Distance at 10 s: x(10)=½·8·10² = 400 m
• Velocity at 5 s: v(5)=8·5 = 40 m/s
• Velocity at 10 s: v(10)=8·10 = 80 m/s

2) Area means distance

• Area 0→5: triangle ½·5·40 = 100 m
• Area 0→10: triangle ½·10·80 = 400 m
• These match the distances above ✓

3) Slope means velocity

• Slope (tangent) at t=5 is about 40 m/s, which matches v(5).

4) Finish line (¼ mile = 400 m)

• Solve ½·8·t² = 400 → 4t²=400 → t=10 s
• Velocity at finish: v=8·10=80 m/s
• In km/h: 80×3.6 = 288 km/h

5) Acceleration check

• Δv from 0→5: a·t = 8·5 = 40 m/s
• Δv from 0→10: 8·10 = 80 m/s
• Meaning: area under a(t) gives change in velocity.

6) (***) Calculus

• x(t)=½·a·t² → dx/dt = v(t)=a·t
• v(t)=a·t → dv/dt = a (constant)
• Integrals: ∫a dt = a·t = v(t), and ∫v(t) dt = ∫a·t dt = ½·a·t² = x(t)

7) (***) Kinetic Energy and Work — Table

Time (s)	Velocity (m/s)	Distance (m)	KE (kJ)	Work (kJ)
2	16	16	128	128
5	40	100	800	800
10	80	400	3200	3200