>A little Estes toy rocket lifts off the pad much more aggressively (in the blink of an eye!) than a full size rocket...

If you really want to, you can reach Mach 10 (~3300 m/s) with a 8 meter long 3500 kg missile in 5 seconds:

https://en.wikipedia.org/wiki/Sprint_(missile)

All of that in the lower atmosphere with the missile heat shield glowing white hot. :)