So, for a Dash 8-100, at 13,000 kg, disregarding drag, engine efficiency, etc, to take-off and climb to 1000m and accelerate to 150 knots (77 m/s), you will need:
- 13000 * 9.81 * 1000 = 127.5 MJ, to reach your altitude
- 0.5 * 13000 * (77)^2 = 38.5 MJ, to accelerate to your climbing speed.
Total: 127 + 38.5 = 166 MJ, or about 46.11 kWh
For a go around, re-accelerating from 1.3 * stall speed (85 knots / 44 m/s) to your climbing speed, and going to your missed approach altitude of 1000 m, you will need:
- 13000 * 9.81 * 1000 = 127.5 MJ, to reach your altitude
- 0.5 * 13000 * (77^2 - 44^2) = 26 MJ to accelerate back to your climbing speed.
Total: 127 + 26 = 153 MJ, or about 42.5 kWh
Nice to see some numbers. So for the peak load situations, a Dash 8-100 would not require a battery bigger than that a short range BEV ("city", though in reality the short range BEV use case is more for the rural equivalent of stuff that would be walkable in a city setting). And that's even before considering the energy contributed by the fossil fuel engine.
"13000 * 9.81 * 1000 = 127.5 MJ, to reach your altitude"
Presumably quite a bit of that would be harvested back during the descent that follows. The conventional engine would still need some excess power (relative to cruise load) to fill the gap left by drag and imperfect circle efficiency of the electric motor/generator, but mass x altitude is stored energy, not lost. (I'm still talking about the "what if we need a second abort" of the root post)