✅ Adventure 4 — Solutions

1) Height vs Time

• y(4)=980−5·16=900 m
• y(10)=980−5·100=480 m
• Distance fallen from 4→10: 900−480=420 m

2) Velocity and Distance

• Speed magnitude: |v(t)|=gt=10t (m/s)
• Area 0→4: triangle ½·4·40=80 m
• Area 0→10: triangle ½·10·100=500 m
• Check with height: 980−y(4)=80 and 980−y(10)=500 ✓

3) Hitting the Ground

• 980−5t²=0 → t²=196 → t=14 s
• Impact speed magnitude: |v(14)|=10·14=140 m/s
• 140×3.6=504 km/h

4) Acceleration Check

• Rectangle area from 4→10: 10·6=60 m/s
• Meaning: area under acceleration = change in velocity. v(10)−v(4)=100−40=60 ✓

★ Calculus (Derivatives + Integrals)

• y(t)=980−5t² → v(t)=−10t (negative means y decreases). Speed = 10t.
• a(t)=v′(t)=−10 (downward). Magnitude is 10.
• Integrate back (using initial conditions): v(t)=∫(−10)dt=−10t, y(t)=∫(−10t)dt=980−5t².