Energy lost per second = 1 J
KE = \(\frac{1}{2}mv^2\)
\(\frac{d(KE)}{dt} = \frac{1}{2}m \times 2v\frac{dv}{dt}\)
\(\frac{d(KE)}{dt} = 1\\ mv \frac{dv}{dt} = 1 \\ vdv = \frac{dt}{m} \\ \int vdv = \frac{1}{m} \int dt \\ \frac{v^2}{2}]_{v_i}^{v_f} = \frac{1}{m}t]_{0}^{t}\)
KE (initial) = 900 = \(\frac{1}{2}mv^2 = 900 = \frac{1}{2}\times 18\times v^2 \\ \Rightarrow v^2 = 100 \Rightarrow v =10ms^{-1}\)
v (initial) = 10m/s
\(v^2_{f} - (100) = \frac{2t}{m}\\ \Rightarrow v_{f}^2 = \frac{2t}{m}+100 \\ v_f = \sqrt{\frac{2t}{m}+100}\)
\(\frac{dx}{dt} = \sqrt{\frac{2t}{18}+100} = \sqrt{\frac{t}{9}+100} = \sqrt{\frac{t+900}{9}} = \frac{\sqrt{t+900}}{3} \\ 3dx =\sqrt{t+900}dt \\ 3\int dx = \int \sqrt{t+900} \, dt \\ 3x = \frac{2}{3} \times (t+900)^{\frac{3}{2}} \\x = \frac{2}{9}\times (t+900)^{\frac{3}{2}}\)
Time took for KE to become zero = 900s
\(x = \frac{2}{9}\times(1800)^\frac{3}{2} =12000\sqrt{2} = 16968m \)
Distance = 16.968 Km (approx 17Km)
