1.Average speed calculation for two equal segments.
Sol: $v_{avg} = \frac{2 v_1 v_2}{v_1 + v_2} = \frac{2(30)(60)}{30 + 60} = \frac{3600}{90} = 40 \text{ km/h}$.
2.Velocity zero time for $x(t) = 3t^2 - 6t + 4$.
Sol: $v(t) = \frac{dx}{dt} = 6t - 6$. Set $v=0 \implies 6t = 6 \implies t = 1 \text{ s}$.
3.Tower height from free fall time $4 \text{ s}$.
Sol: $h = \frac{1}{2}gt^2 = \frac{1}{2}(9.8)(4^2) = 4.9 \times 16 = 78.4 \text{ m}$.
Sol: A straight line starting from positive $v_0$ sloping downwards, crossing time axis at $t_0$, and extending continuously into the negative velocity region with uniform slope $-a$.
5.Time for train to cross bridge.
Sol: Total distance $D = 150 + 850 = 1000 \text{ m} = 1 \text{ km}$.
Speed $v = 45 \text{ km/h} = 45 \times \frac{5}{18} = 12.5 \text{ m/s}$.
Time $t = \frac{1000}{12.5} = 80 \text{ s}$.
